diff --git a/.gitignore b/.gitignore
--- a/.gitignore
+++ b/.gitignore
@@ -1,4 +1,5 @@
 dist
+dist-newstyle
 docs
 wiki
 TAGS
@@ -11,3 +12,21 @@
 *.hi
 *~
 *#
+.stack-work/
+cabal-dev
+*.chi
+*.chs.h
+*.dyn_o
+*.dyn_hi
+.hpc
+.hsenv
+.cabal-sandbox/
+cabal.sandbox.config
+*.prof
+*.aux
+*.hp
+*.eventlog
+cabal.project.local
+cabal.project.local~
+.HTF/
+.ghc.environment.*
diff --git a/.travis.yml b/.travis.yml
--- a/.travis.yml
+++ b/.travis.yml
@@ -1,108 +1,144 @@
-# This file has been generated -- see https://github.com/hvr/multi-ghc-travis
+# This Travis job script has been generated by a script via
+#
+#   runghc make_travis_yml_2.hs '-o' '.travis.yml' '--irc-channel=irc.freenode.org#haskell-lens' '--no-no-tests-no-bench' '--no-installed' 'cabal.project'
+#
+# For more information, see https://github.com/hvr/multi-ghc-travis
+#
 language: c
 sudo: false
 
+git:
+  submodules: false  # whether to recursively clone submodules
+
+notifications:
+  irc:
+    channels:
+      - "irc.freenode.org#haskell-lens"
+    skip_join: true
+    template:
+      - "\x0313kan-extensions\x03/\x0306%{branch}\x03 \x0314%{commit}\x03 %{build_url} %{message}"
+
 cache:
   directories:
-    - $HOME/.cabsnap
     - $HOME/.cabal/packages
+    - $HOME/.cabal/store
 
 before_cache:
   - rm -fv $HOME/.cabal/packages/hackage.haskell.org/build-reports.log
-  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/00-index.tar
+  # remove files that are regenerated by 'cabal update'
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/00-index.*
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/*.json
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.cache
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.tar
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.tar.idx
 
+  - rm -rfv $HOME/.cabal/packages/head.hackage
+
+addons:
+  apt:
+    packages: &apt_packages
+      - ghc-ppa-tools
+      - alex-3.1.7
+      - happy-1.19.5
+
 matrix:
   include:
-    - env: CABALVER=1.18 GHCVER=7.4.2
-      compiler: ": #GHC 7.4.2"
-      addons: {apt: {packages: [cabal-install-1.18,ghc-7.4.2,alex-3.1.4,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.18 GHCVER=7.6.3
-      compiler: ": #GHC 7.6.3"
-      addons: {apt: {packages: [cabal-install-1.18,ghc-7.6.3,alex-3.1.4,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.18 GHCVER=7.8.4
-      compiler: ": #GHC 7.8.4"
-      addons: {apt: {packages: [cabal-install-1.18,ghc-7.8.4,alex-3.1.4,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.22 GHCVER=7.10.1
-      compiler: ": #GHC 7.10.1"
-      addons: {apt: {packages: [cabal-install-1.22,ghc-7.10.1,alex-3.1.4,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.22 GHCVER=7.10.3
-      compiler: ": #GHC 7.10.3"
-      addons: {apt: {packages: [cabal-install-1.22,ghc-7.10.3,alex-3.1.4,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.24 GHCVER=8.0.2
-      compiler: ": #GHC 8.0.2"
-      addons: {apt: {packages: [cabal-install-1.24,ghc-8.0.2,alex-3.1.4,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=2.0 GHCVER=8.2.1
-      compiler: ": #GHC 8.2.1"
-      addons: {apt: {packages: [cabal-install-2.0,ghc-8.2.1,alex-3.1.4,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=head GHCVER=head
-      compiler: ": #GHC head"
-      addons: {apt: {packages: [cabal-install-head,ghc-head,alex-3.1.4,happy-1.19.5], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.4.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.4.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.6.3"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.6.3], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.8.4"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.8.4], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.10.3"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.10.3], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.0.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-8.0.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.2.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-8.2.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.4.1"
+      env: GHCHEAD=true
+      addons: {apt: {packages: [*apt_packages,cabal-install-head,ghc-8.4.1], sources: [hvr-ghc]}}
+    - compiler: "ghc-head"
+      env: GHCHEAD=true
+      addons: {apt: {packages: [*apt_packages,cabal-install-head,ghc-head], sources: [hvr-ghc]}}
 
   allow_failures:
-  - env: CABALVER=head GHCVER=head
-
+    - compiler: "ghc-8.4.1"
+    - compiler: "ghc-head"
 before_install:
- - unset CC
- - export HAPPYVER=1.19.5
- - export ALEXVER=3.1.4
- - export PATH=~/.cabal/bin:/opt/ghc/$GHCVER/bin:/opt/cabal/$CABALVER/bin:/opt/happy/$HAPPYVER/bin:/opt/alex/$ALEXVER/bin:$PATH
+  - HC=${CC}
+  - HCPKG=${HC/ghc/ghc-pkg}
+  - unset CC
+  - ROOTDIR=$(pwd)
+  - mkdir -p $HOME/.local/bin
+  - "PATH=/opt/ghc/bin:/opt/ghc-ppa-tools/bin:$HOME/local/bin:$PATH"
+  - HCNUMVER=$(( $(${HC} --numeric-version|sed -E 's/([0-9]+)\.([0-9]+)\.([0-9]+).*/\1 * 10000 + \2 * 100 + \3/') ))
+  - echo $HCNUMVER
 
 install:
- - cabal --version
- - echo "$(ghc --version) [$(ghc --print-project-git-commit-id 2> /dev/null || echo '?')]"
- - if [ -f $HOME/.cabal/packages/hackage.haskell.org/00-index.tar.gz ];
-   then
-     zcat $HOME/.cabal/packages/hackage.haskell.org/00-index.tar.gz >
-          $HOME/.cabal/packages/hackage.haskell.org/00-index.tar;
-   fi
- - travis_retry cabal update
- - "sed -i  's/^jobs:.*$/jobs: 2/' $HOME/.cabal/config"
- - cabal install --only-dependencies --enable-tests --enable-benchmarks --dry -v > installplan.txt
- - sed -i -e '1,/^Resolving /d' installplan.txt; cat installplan.txt
+  - cabal --version
+  - echo "$(${HC} --version) [$(${HC} --print-project-git-commit-id 2> /dev/null || echo '?')]"
+  - BENCH=${BENCH---enable-benchmarks}
+  - TEST=${TEST---enable-tests}
+  - HADDOCK=${HADDOCK-true}
+  - INSTALLED=${INSTALLED-true}
+  - GHCHEAD=${GHCHEAD-false}
+  - travis_retry cabal update -v
+  - "sed -i.bak 's/^jobs:/-- jobs:/' ${HOME}/.cabal/config"
+  - rm -fv cabal.project cabal.project.local
+  # Overlay Hackage Package Index for GHC HEAD: https://github.com/hvr/head.hackage
+  - |
+    if $GHCHEAD; then
+      sed -i.bak 's/-- allow-newer:.*/allow-newer: *:base, *:template-haskell, *:ghc, *:Cabal/' ${HOME}/.cabal/config
 
-# check whether current requested install-plan matches cached package-db snapshot
- - if diff -u installplan.txt $HOME/.cabsnap/installplan.txt;
-   then
-     echo "cabal build-cache HIT";
-     rm -rfv .ghc;
-     cp -a $HOME/.cabsnap/ghc $HOME/.ghc;
-     cp -a $HOME/.cabsnap/lib $HOME/.cabsnap/share $HOME/.cabsnap/bin $HOME/.cabal/;
-   else
-     echo "cabal build-cache MISS";
-     rm -rf $HOME/.cabsnap;
-     mkdir -p $HOME/.ghc $HOME/.cabal/lib $HOME/.cabal/share $HOME/.cabal/bin;
-     cabal install --only-dependencies --enable-tests --enable-benchmarks;
-   fi
+      echo 'repository head.hackage'                                                        >> ${HOME}/.cabal/config
+      echo '   url: http://head.hackage.haskell.org/'                                       >> ${HOME}/.cabal/config
+      echo '   secure: True'                                                                >> ${HOME}/.cabal/config
+      echo '   root-keys: 07c59cb65787dedfaef5bd5f987ceb5f7e5ebf88b904bbd4c5cbdeb2ff71b740' >> ${HOME}/.cabal/config
+      echo '              2e8555dde16ebd8df076f1a8ef13b8f14c66bad8eafefd7d9e37d0ed711821fb' >> ${HOME}/.cabal/config
+      echo '              8f79fd2389ab2967354407ec852cbe73f2e8635793ac446d09461ffb99527f6e' >> ${HOME}/.cabal/config
+      echo '   key-threshold: 3'                                                            >> ${HOME}/.cabal.config
 
-# snapshot package-db on cache miss
- - if [ ! -d $HOME/.cabsnap ];
-   then
-      echo "snapshotting package-db to build-cache";
-      mkdir $HOME/.cabsnap;
-      cp -a $HOME/.ghc $HOME/.cabsnap/ghc;
-      cp -a $HOME/.cabal/lib $HOME/.cabal/share $HOME/.cabal/bin installplan.txt $HOME/.cabsnap/;
-   fi
+      cabal new-update head.hackage -v
+    fi
+  - grep -Ev -- '^\s*--' ${HOME}/.cabal/config | grep -Ev '^\s*$'
+  - "printf 'packages: \".\"\\n' > cabal.project"
+  - cat cabal.project
+  - if [ -f "./configure.ac" ]; then
+      (cd "." && autoreconf -i);
+    fi
+  - rm -f cabal.project.freeze
+  - cabal new-build -w ${HC} ${TEST} ${BENCH} --project-file="cabal.project" --dep -j2 all
+  - rm -rf "."/.ghc.environment.* "."/dist
+  - DISTDIR=$(mktemp -d /tmp/dist-test.XXXX)
 
 # Here starts the actual work to be performed for the package under test;
 # any command which exits with a non-zero exit code causes the build to fail.
 script:
- - cabal configure --enable-tests -v2  # -v2 provides useful information for debugging
- - cabal build   # this builds all libraries and executables (including tests)
- - cabal test
- - cabal bench || true  # expected result: these will crash
- - cabal sdist || true  # tests that a source-distribution can be generated
+  # test that source-distributions can be generated
+  - (cd "." && cabal sdist)
+  - mv "."/dist/kan-extensions-*.tar.gz ${DISTDIR}/
+  - cd ${DISTDIR} || false
+  - find . -maxdepth 1 -name '*.tar.gz' -exec tar -xvf '{}' \;
+  - "printf 'packages: kan-extensions-*/*.cabal\\n' > cabal.project"
+  - cat cabal.project
 
-# Check that the resulting source distribution can be built & installed.
-# If there are no other `.tar.gz` files in `dist`, this can be even simpler:
-# `cabal install --force-reinstalls dist/*-*.tar.gz`
- - SRC_TGZ=$(cabal info . | awk '{print $2;exit}').tar.gz &&
-   (cd dist && cabal install --force-reinstalls "$SRC_TGZ")
 
-notifications:
-  irc:
-    channels:
-      - "irc.freenode.org#haskell-lens"
-    skip_join: true
-    template:
-      - "\x0313kan-extensions\x0f/\x0306%{branch}\x0f \x0314%{commit}\x0f %{message} \x0302\x1f%{build_url}\x0f"
+  # build & run tests, build benchmarks
+  - cabal new-build -w ${HC} ${TEST} ${BENCH} all
+
+  # cabal check
+  - (cd kan-extensions-* && cabal check)
+
+  # haddock
+  - rm -rf ./dist-newstyle
+  - if $HADDOCK; then cabal new-haddock -w ${HC} ${TEST} ${BENCH} all; else echo "Skipping haddock generation";fi
+
+# REGENDATA ["-o",".travis.yml","--irc-channel=irc.freenode.org#haskell-lens","--no-no-tests-no-bench","--no-installed","cabal.project"]
 # EOF
diff --git a/CHANGELOG.markdown b/CHANGELOG.markdown
--- a/CHANGELOG.markdown
+++ b/CHANGELOG.markdown
@@ -1,3 +1,12 @@
+5.1 [2018.01.28]
+----------------
+* Make `Density`, `Codensity`, `Kan` and `Lan` polykinded.
+* Add `Eq1`, `Ord1`, `Read1` and `Show1` instances for `Coyoneda` and `Yoneda`.
+* Change contexts of `Eq` and `Ord` instances of `Coyoneda` and `Yoneda`
+  (and the `Show` instance for `Coyoneda`) to require lifted class instances,
+  e.g. `Eq1 f, Eq a`.
+* Allow `free-5`.
+
 5.0.2
 -----
 * Added `hoistCoyoneda`
diff --git a/include/kan-extensions-common.h b/include/kan-extensions-common.h
new file mode 100644
--- /dev/null
+++ b/include/kan-extensions-common.h
@@ -0,0 +1,19 @@
+#ifndef MIN_VERSION_base
+#define MIN_VERSION_base(x,y,z) 1
+#endif
+
+#ifndef MIN_VERSION_mtl
+#define MIN_VERSION_mtl(x,y,z) 1
+#endif
+
+#if MIN_VERSION_base(4,9,0)
+#define LIFTED_FUNCTOR_CLASSES 1
+#else
+#if MIN_VERSION_transformers(0,5,0)
+#define LIFTED_FUNCTOR_CLASSES 1
+#else
+#if MIN_VERSION_transformers_compat(0,5,0) && !MIN_VERSION_transformers(0,4,0)
+#define LIFTED_FUNCTOR_CLASSES 1
+#endif
+#endif
+#endif
diff --git a/kan-extensions.cabal b/kan-extensions.cabal
--- a/kan-extensions.cabal
+++ b/kan-extensions.cabal
@@ -1,6 +1,6 @@
 name:          kan-extensions
 category:      Data Structures, Monads, Comonads, Functors
-version:       5.0.2
+version:       5.1
 license:       BSD3
 cabal-version: >= 1.6
 license-file:  LICENSE
@@ -10,9 +10,16 @@
 homepage:      http://github.com/ekmett/kan-extensions/
 bug-reports:   http://github.com/ekmett/kan-extensions/issues
 copyright:     Copyright (C) 2008-2016 Edward A. Kmett
-synopsis:      Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads
-description:   Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads
+synopsis:      Kan extensions, Kan lifts, the Yoneda lemma, and (co)density (co)monads
+description:   Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads.
 build-type:    Simple
+tested-with:   GHC == 7.4.2
+             , GHC == 7.6.3
+             , GHC == 7.8.4
+             , GHC == 7.10.3
+             , GHC == 8.0.2
+             , GHC == 8.2.2
+             , GHC == 8.4.1
 
 extra-source-files:
   .travis.yml
@@ -21,6 +28,7 @@
   .vim.custom
   CHANGELOG.markdown
   README.markdown
+  include/kan-extensions-common.h
 
 source-repository head
   type: git
@@ -28,6 +36,8 @@
 
 library
   hs-source-dirs: src
+  include-dirs: include
+  includes: kan-extensions-common.h
 
   other-extensions:
     CPP
@@ -48,11 +58,13 @@
     contravariant >= 1       && < 2,
     distributive  >= 0.2.2   && < 1,
     fail          >= 4.9     && < 5,
-    free          >= 4       && < 5,
+    free          >= 4       && < 6,
     mtl           >= 2.0.1   && < 2.3,
+    profunctors   >= 5       && < 6,
     semigroupoids >= 4       && < 6,
     tagged        >= 0.7.2   && < 1,
-    transformers  >= 0.2     && < 0.6
+    transformers  >= 0.2     && < 0.6,
+    transformers-compat >= 0.3 && <0.6
 
   exposed-modules:
     Control.Comonad.Density
@@ -70,3 +82,6 @@
 
   ghc-options: -Wall
 
+  -- See https://ghc.haskell.org/trac/ghc/wiki/Migration/8.0#base-4.9.0.0
+  if impl(ghc >= 8.0)
+    ghc-options: -Wcompat -Wnoncanonical-monad-instances -Wnoncanonical-monadfail-instances
diff --git a/src/Control/Comonad/Density.hs b/src/Control/Comonad/Density.hs
--- a/src/Control/Comonad/Density.hs
+++ b/src/Control/Comonad/Density.hs
@@ -1,5 +1,8 @@
 {-# LANGUAGE CPP #-}
 {-# LANGUAGE GADTs #-}
+#if __GLASGOW_HASKELL__ >= 706
+{-# LANGUAGE PolyKinds #-}
+#endif
 {-# LANGUAGE MultiParamTypeClasses #-}
 #if __GLASGOW_HASKELL__ >= 702 && __GLASGOW_HASKELL__ < 710
 {-# LANGUAGE Trustworthy #-}
diff --git a/src/Control/Monad/Co.hs b/src/Control/Monad/Co.hs
--- a/src/Control/Monad/Co.hs
+++ b/src/Control/Monad/Co.hs
@@ -4,6 +4,9 @@
 {-# LANGUAGE FlexibleContexts #-}
 {-# LANGUAGE UndecidableInstances #-}
 {-# LANGUAGE MultiParamTypeClasses #-}
+#if __GLASGOW_HASKELL__ >= 706
+{-# LANGUAGE PolyKinds #-}
+#endif
 #if __GLASGOW_HASKELL__ >= 702 && __GLASGOW_HASKELL__ < 710
 {-# LANGUAGE Trustworthy #-}
 #endif
@@ -29,7 +32,7 @@
 -- which is a 'Monad' in @Hask^op@ to a 'Monad' in 'Hask'.
 --
 -- For any @r@, the 'Contravariant' functor / presheaf @(-> r)@ :: Hask^op -> Hask is adjoint to the \"same\"
--- 'Contravariant' functor @(-> r) :: Hask -> Hask^op@. So we can sandwhich a
+-- 'Contravariant' functor @(-> r) :: Hask -> Hask^op@. So we can sandwich a
 -- Monad in Hask^op in the middle to obtain @w (a -> r-) -> r+@, and then take a coend over
 -- @r@ to obtain @forall r. w (a -> r) -> r@. This gives rise to 'Co'. If we observe that
 -- we didn't care what the choices we made for @r@ were to finish this construction, we can
diff --git a/src/Control/Monad/Codensity.hs b/src/Control/Monad/Codensity.hs
--- a/src/Control/Monad/Codensity.hs
+++ b/src/Control/Monad/Codensity.hs
@@ -3,6 +3,9 @@
 {-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE UndecidableInstances #-}
 {-# LANGUAGE MultiParamTypeClasses #-}
+#if __GLASGOW_HASKELL__ >= 706
+{-# LANGUAGE PolyKinds #-}
+#endif
 #if __GLASGOW_HASKELL__ >= 702
 {-# LANGUAGE Trustworthy #-}
 #endif
@@ -57,7 +60,7 @@
 -- See \"Asymptotic Improvement of Computations over Free Monads\" by Janis
 -- Voigtländer for more information about this type.
 --
--- <http://www.iai.uni-bonn.de/~jv/mpc08.pdf>
+-- <https://www.janis-voigtlaender.eu/papers/AsymptoticImprovementOfComputationsOverFreeMonads.pdf>
 newtype Codensity m a = Codensity
   { runCodensity :: forall b. (a -> m b) -> m b
   }
diff --git a/src/Data/Functor/Coyoneda.hs b/src/Data/Functor/Coyoneda.hs
--- a/src/Data/Functor/Coyoneda.hs
+++ b/src/Data/Functor/Coyoneda.hs
@@ -9,6 +9,7 @@
 #if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 702
 {-# LANGUAGE Trustworthy #-}
 #endif
+#include "kan-extensions-common.h"
 
 -----------------------------------------------------------------------------
 -- |
@@ -36,9 +37,12 @@
 import Control.Comonad
 import Control.Comonad.Trans.Class
 import Data.Distributive
+#if !LIFTED_FUNCTOR_CLASSES
 import Data.Function (on)
+#endif
 import Data.Functor.Adjunction
 import Data.Functor.Bind
+import Data.Functor.Classes
 import Data.Functor.Extend
 import Data.Functor.Identity
 import Data.Functor.Kan.Lan
@@ -145,13 +149,19 @@
 
 instance Monad m => Monad (Coyoneda m) where
 #if __GLASGOW_HASKELL__ < 710
+  -- pre-AMP
   return = Coyoneda id . return
   {-# INLINE return #-}
+
+  Coyoneda _ m >> Coyoneda g n = Coyoneda g (m >> n)
+  {-# INLINE (>>) #-}
+#else
+  -- post-AMP
+  (>>) = (*>)
+  {-# INLINE (>>) #-}
 #endif
   Coyoneda f v >>= k = lift (v >>= lowerM . k . f)
   {-# INLINE (>>=) #-}
-  Coyoneda _ m >> Coyoneda g n = Coyoneda g (m >> n)
-  {-# INLINE (>>) #-}
 
 instance MonadTrans Coyoneda where
   lift = Coyoneda id
@@ -208,25 +218,74 @@
   collect f = liftCoyoneda . collect (lowerCoyoneda . f)
   {-# INLINE collect #-}
 
-instance (Functor f, Show (f a)) => Show (Coyoneda f a) where
-  showsPrec d (Coyoneda f a) = showParen (d > 10) $
-    showString "liftCoyoneda " . showsPrec 11 (fmap f a)
+instance (Functor f, Show1 f) => Show1 (Coyoneda f) where
+#if LIFTED_FUNCTOR_CLASSES
+  liftShowsPrec sp sl d (Coyoneda f a) =
+    showsUnaryWith (liftShowsPrec sp sl) "liftCoyoneda" d (fmap f a)
+  {-# INLINE liftShowsPrec #-}
+#else
+  showsPrec1 d (Coyoneda f a) = showParen (d > 10) $
+    showString "liftCoyoneda " . showsPrec1 11 (fmap f a)
+  {-# INLINE showsPrec1 #-}
+#endif
+
+instance (Read1 f) => Read1 (Coyoneda f) where
+#if LIFTED_FUNCTOR_CLASSES
+  liftReadsPrec rp rl = readsData $
+    readsUnaryWith (liftReadsPrec rp rl) "liftCoyoneda" liftCoyoneda
+  {-# INLINE liftReadsPrec #-}
+#else
+  readsPrec1 d = readParen (d > 10) $ \r' ->
+    [ (liftCoyoneda f, t)
+    | ("liftCoyoneda", s) <- lex r'
+    , (f, t) <- readsPrec1 11 s
+    ]
+  {-# INLINE readsPrec1 #-}
+#endif
+
+instance (Functor f, Show1 f, Show a) => Show (Coyoneda f a) where
+  showsPrec = showsPrec1
   {-# INLINE showsPrec #-}
 
-#ifdef __GLASGOW_HASKELL__
 instance Read (f a) => Read (Coyoneda f a) where
+#ifdef __GLASGOW_HASKELL__
   readPrec = parens $ prec 10 $ do
     Ident "liftCoyoneda" <- lexP
     liftCoyoneda <$> step readPrec
   {-# INLINE readPrec #-}
+#else
+  readsPrec d = readParen (d > 10) $ \r' ->
+    [ (liftCoyoneda f, t)
+    | ("liftCoyoneda", s) <- lex r'
+    , (f, t) <- readsPrec 11 s
+    ]
+  {-# INLINE readsPrec #-}
 #endif
 
-instance (Functor f, Eq (f a)) => Eq (Coyoneda f a) where
-  (==) = (==) `on` lowerCoyoneda
+instance (Functor f, Eq1 f) => Eq1 (Coyoneda f) where
+#if LIFTED_FUNCTOR_CLASSES
+  liftEq eq = \x y -> liftEq eq (lowerCoyoneda x) (lowerCoyoneda y)
+  {-# INLINE liftEq #-}
+#else
+  eq1 = eq1 `on` lowerCoyoneda
+  {-# INLINE eq1 #-}
+#endif
+
+instance (Functor f, Ord1 f) => Ord1 (Coyoneda f) where
+#if LIFTED_FUNCTOR_CLASSES
+  liftCompare cmp = \x y -> liftCompare cmp (lowerCoyoneda x) (lowerCoyoneda y)
+  {-# INLINE liftCompare #-}
+#else
+  compare1 = compare1 `on` lowerCoyoneda
+  {-# INLINE compare1 #-}
+#endif
+
+instance (Functor f, Eq1 f, Eq a) => Eq (Coyoneda f a) where
+  (==) = eq1
   {-# INLINE (==) #-}
 
-instance (Functor f, Ord (f a)) => Ord (Coyoneda f a) where
-  compare = compare `on` lowerCoyoneda
+instance (Functor f, Ord1 f, Ord a) => Ord (Coyoneda f a) where
+  compare = compare1
   {-# INLINE compare #-}
 
 instance Adjunction f g => Adjunction (Coyoneda f) (Coyoneda g) where
diff --git a/src/Data/Functor/Day.hs b/src/Data/Functor/Day.hs
--- a/src/Data/Functor/Day.hs
+++ b/src/Data/Functor/Day.hs
@@ -35,21 +35,24 @@
   , intro1, intro2
   , elim1, elim2
   , trans1, trans2
+  , cayley, dayley
   ) where
 
 import Control.Applicative
+import Control.Category
 import Control.Comonad
 import Control.Comonad.Trans.Class
 import Data.Distributive
+import Data.Profunctor.Cayley (Cayley(..))
+import Data.Profunctor.Composition (Procompose(..))
 import Data.Functor.Identity
 import Data.Functor.Rep
 #ifdef __GLASGOW_HASKELL__
 import Data.Typeable
 #endif
+import Prelude hiding (id,(.))
 
 -- | The Day convolution of two covariant functors.
---
--- @Day f g a -> h a@ is isomorphic to @f a -> Rift g h a@
 data Day f g a = forall b c. Day (f b) (g c) (b -> c -> a)
 #if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 707
   deriving Typeable
@@ -204,3 +207,10 @@
 -- @
 trans2 :: (forall x. g x -> h x) -> Day f g a -> Day f h a
 trans2 gh (Day fb gc bca) = Day fb (gh gc) bca
+
+cayley :: Procompose (Cayley f p) (Cayley g q) a b -> Cayley (Day f g) (Procompose p q) a b
+cayley (Procompose (Cayley p) (Cayley q)) = Cayley $ Day p q Procompose
+
+-- | Proposition 4.1 from Pastro and Street
+dayley :: Category p => Procompose (Cayley f p) (Cayley g p) a b -> Cayley (Day f g) p a b
+dayley (Procompose (Cayley p) (Cayley q)) = Cayley $ Day p q (.)
diff --git a/src/Data/Functor/Kan/Lan.hs b/src/Data/Functor/Kan/Lan.hs
--- a/src/Data/Functor/Kan/Lan.hs
+++ b/src/Data/Functor/Kan/Lan.hs
@@ -1,6 +1,9 @@
 {-# LANGUAGE CPP #-}
 {-# LANGUAGE GADTs #-}
 {-# LANGUAGE Rank2Types #-}
+#if __GLASGOW_HASKELL__ >= 706
+{-# LANGUAGE PolyKinds #-}
+#endif
 #if __GLASGOW_HASKELL__ >= 702 && __GLASGOW_HASKELL__ < 710
 {-# LANGUAGE Trustworthy #-}
 #endif
diff --git a/src/Data/Functor/Kan/Ran.hs b/src/Data/Functor/Kan/Ran.hs
--- a/src/Data/Functor/Kan/Ran.hs
+++ b/src/Data/Functor/Kan/Ran.hs
@@ -1,5 +1,8 @@
 {-# LANGUAGE Rank2Types, GADTs #-}
 {-# LANGUAGE CPP #-}
+#if __GLASGOW_HASKELL__ >= 706
+{-# LANGUAGE PolyKinds #-}
+#endif
 #if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 702
 {-# LANGUAGE Trustworthy #-}
 #endif
@@ -38,7 +41,7 @@
 -- from the universal property.
 --
 -- Given a 'Functor' @h : C -> D@ and a 'Functor' @g : C -> C'@, we want to extend @h@ /back/ along @g@
--- to give @Ran g h : C' -> C@, such that the natural transformation @'gran' :: Ran g h (g a) -> h a@ exists.
+-- to give @Ran g h : C' -> D@, such that the natural transformation @'gran' :: Ran g h (g a) -> h a@ exists.
 --
 -- In some sense this is trying to approximate the inverse of @g@ by using one of
 -- its adjoints, because if the adjoint and the inverse both exist, they match!
@@ -52,7 +55,7 @@
 --
 -- The Right Kan extension is unique (up to isomorphism) by taking this as its universal property.
 --
--- That is to say given any @K : C' -> C@ such that we have a natural transformation from @k.g@ to @h@
+-- That is to say given any @K : C' -> D@ such that we have a natural transformation from @k.g@ to @h@
 -- @(forall x. k (g x) -> h x)@ there exists a canonical natural transformation from @k@ to @Ran g h@.
 -- @(forall x. k x -> Ran g h x)@.
 --
diff --git a/src/Data/Functor/Yoneda.hs b/src/Data/Functor/Yoneda.hs
--- a/src/Data/Functor/Yoneda.hs
+++ b/src/Data/Functor/Yoneda.hs
@@ -8,6 +8,7 @@
 #if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 702
 {-# LANGUAGE Trustworthy #-}
 #endif
+#include "kan-extensions-common.h"
 
 -----------------------------------------------------------------------------
 -- |
@@ -44,9 +45,9 @@
 import Control.Comonad.Trans.Class
 import Data.Distributive
 import Data.Foldable
-import Data.Function (on)
 import Data.Functor.Adjunction
 import Data.Functor.Bind
+import Data.Functor.Classes
 import Data.Functor.Extend
 import Data.Functor.Identity
 import Data.Functor.Kan.Ran
@@ -138,24 +139,67 @@
   unit = liftYoneda . fmap liftYoneda . unit
   counit (Yoneda m) = counit (m lowerYoneda)
 
--- instance Show1 f => Show1 (Yoneda f) where
+instance Show1 f => Show1 (Yoneda f) where
+#if LIFTED_FUNCTOR_CLASSES
+  liftShowsPrec sp sl d (Yoneda f) =
+    showsUnaryWith (liftShowsPrec sp sl) "liftYoneda" d (f id)
+#else
+  showsPrec1 d (Yoneda f) = showParen (d > 10) $
+    showString "liftYoneda " . showsPrec1 11 (f id)
+#endif
+
+instance (Read1 f, Functor f) => Read1 (Yoneda f) where
+#if LIFTED_FUNCTOR_CLASSES
+  liftReadsPrec rp rl = readsData $
+    readsUnaryWith (liftReadsPrec rp rl) "liftYoneda" liftYoneda
+#else
+  readsPrec1 d = readParen (d > 10) $ \r' ->
+    [ (liftYoneda f, t)
+    | ("liftYoneda", s) <- lex r'
+    , (f, t) <- readsPrec1 11 s
+    ]
+#endif
+
 instance Show (f a) => Show (Yoneda f a) where
   showsPrec d (Yoneda f) = showParen (d > 10) $
     showString "liftYoneda " . showsPrec 11 (f id)
 
--- instance Read1 f => Read1 (Yoneda f) where
-#ifdef __GLASGOW_HASKELL__
 instance (Functor f, Read (f a)) => Read (Yoneda f a) where
+#ifdef __GLASGOW_HASKELL__
   readPrec = parens $ prec 10 $ do
      Ident "liftYoneda" <- lexP
      liftYoneda <$> step readPrec
+#else
+  readsPrec d = readParen (d > 10) $ \r' ->
+    [ (liftYoneda f, t)
+    | ("liftYoneda", s) <- lex r'
+    , (f, t) <- readsPrec 11 s
+    ]
 #endif
 
-instance Eq (f a) => Eq (Yoneda f a) where
-  (==) = (==) `on` lowerYoneda
+infixl 0 `on1`
+on1 :: (g a -> g b -> c) -> (forall x. f x -> g x) -> f a -> f b -> c
+(.*.) `on1` f = \x y -> f x .*. f y
 
-instance Ord (f a) => Ord (Yoneda f a) where
-  compare = compare `on` lowerYoneda
+instance Eq1 f => Eq1 (Yoneda f) where
+#if LIFTED_FUNCTOR_CLASSES
+  liftEq eq = liftEq eq `on1` lowerYoneda
+#else
+  eq1 = eq1 `on1` lowerYoneda
+#endif
+
+instance Ord1 f => Ord1 (Yoneda f) where
+#if LIFTED_FUNCTOR_CLASSES
+  liftCompare cmp = liftCompare cmp `on1` lowerYoneda
+#else
+  compare1 = compare1 `on1` lowerYoneda
+#endif
+
+instance (Eq1 f, Eq a) => Eq (Yoneda f a) where
+  (==) = eq1
+
+instance (Ord1 f, Ord a) => Ord (Yoneda f a) where
+  compare = compare1
 
 maxF :: (Functor f, Ord (f a)) => Yoneda f a -> Yoneda f a -> Yoneda f a
 Yoneda f `maxF` Yoneda g = liftYoneda $ f id `max` g id