diff --git a/CHANGELOG.md b/CHANGELOG.md
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,3 +1,9 @@
+# 0.7.0.0
+
+- Changed `Monoid` instance for `Generically`, to be compatible with users'
+  non-generic instances of `Semigroup`. Thanks to yairchu.
+- Add `gcoerce`, `gcoerceBinop`.
+
 # 0.6.0.1
 
 - Fix derivation of `Show1` for `(:.:)`
diff --git a/README.md b/README.md
--- a/README.md
+++ b/README.md
@@ -4,7 +4,34 @@
 
 ## Generic deriving for standard classes
 
+### Example: generically deriving Semigroup instances for products
+
+Semi-automatic method using `gmappend`
+
 ```haskell
+data Foo a = Bar [a] [a] deriving Generic
+
+instance Semigroup (Foo a) where
+  (<>) = gmappend
+```
+
+This library also synergizes with the `DerivingVia` extension
+(introduced in GHC 8.6), thanks to the `Generically` newtype.
+
+```haskell
+data Foo a = Bar [a] [a]
+  deriving Generic
+  deriving Semigroup via (Generically (Foo a))
+```
+
+These examples can be found in `test/example.hs`.
+
+---
+
+Note for completeness, the first example uses the following extensions and
+imports:
+
+```haskell
 {-# LANGUAGE DeriveGeneric #-}
 
 -- base
@@ -12,35 +39,32 @@
 import GHC.Generics
 
 -- generic-data
-import Generic.Data (gmappend, Generically(..))
+import Generic.Data (gmappend)
 import Generic.Data.Orphans ()
-
-data Foo a = Bar [a] [a] deriving Generic
-
-instance Semigroup (Foo a) where
-  (<>) = gmappend
+```
 
--- also with some additional extensions --
+The second example makes these additions on top:
 
+```haskell
 {-# LANGUAGE
     DerivingStrategies,
     DerivingVia #-}  -- since GHC 8.6.1
 
-data Foo a = Bar [a] [a]
-  deriving Generic
-  deriving Semigroup via (Generically (Foo a))
-
--- This example can be found in test/example.hs
+-- In addition to the previous imports
+import Generic.Data (Generically(..))
 ```
 
+### Supported classes
+
 Supported classes that GHC currently can't derive: `Semigroup`, `Monoid`,
 `Applicative`, `Alternative`, `Eq1`, `Ord1`, `Show1`.
 
 Other classes from base are also supported, even though GHC can already derive
 them:
 
-- `Eq`, `Ord`, `Enum`, `Bounded`, `Show` (standard);
-- `Functor`, `Foldable`, `Traversable` (via extensions, `DeriveFunctor`, etc.).
+- `Eq`, `Ord`, `Enum`, `Bounded`, `Show` (derivable by the standard);
+- `Functor`, `Foldable`, `Traversable` (derivable via extensions,
+  `DeriveFunctor`, etc.).
 
 (`Read` is currently not implemented.)
 
@@ -75,12 +99,11 @@
 
 ```haskell
 {-# LANGUAGE DeriveGeneric #-}
-
 import GHC.Generic (Generic)
-
 import Generic.Data (gshowsPrec)
 import Generic.Data.Microsurgery (toData, derecordify)
 
+-- An example record type
 newtype T = T { unT :: Int } deriving Generic
 
 -- Naively deriving Show would result in this being shown:
@@ -102,7 +125,6 @@
 
 ```haskell
 {-# LANGUAGE DeriveGeneric, DerivingVia #-}
-
 import GHC.Generic (Generic)
 
 -- Constructors must be visible to use DerivingVia
diff --git a/generic-data.cabal b/generic-data.cabal
--- a/generic-data.cabal
+++ b/generic-data.cabal
@@ -1,5 +1,5 @@
 name:                generic-data
-version:             0.6.0.1
+version:             0.7.0.0
 synopsis:            Deriving instances with GHC.Generics and related utilities
 description:
   Generic implementations of standard type classes.
diff --git a/src/Generic/Data.hs b/src/Generic/Data.hs
--- a/src/Generic/Data.hs
+++ b/src/Generic/Data.hs
@@ -113,6 +113,10 @@
   , pack
   , unpack
 
+    -- * Generic coercions
+  , gcoerce
+  , gcoerceBinop
+
     -- * Accessing metadata
 
     -- | Using @TypeApplications@.
@@ -163,3 +167,4 @@
 import Generic.Data.Internal.Show
 import Generic.Data.Internal.Newtype
 import Generic.Data.Internal.Resolvers
+import Generic.Data.Internal.Utils
diff --git a/src/Generic/Data/Internal/Data.hs b/src/Generic/Data/Internal/Data.hs
--- a/src/Generic/Data/Internal/Data.hs
+++ b/src/Generic/Data/Internal/Data.hs
@@ -31,11 +31,11 @@
            , Eq, Ord, Eq1, Ord1, Semigroup, Monoid )
 
 -- | Conversion between a generic type and the synthetic type made using its
--- representation.
+-- representation. Inverse of 'fromData'.
 toData :: Generic a => a -> Data (Rep a) p
 toData = Data . from
 
--- | Inverse of 'fromData'.
+-- | Inverse of 'toData'.
 fromData :: Generic a => Data (Rep a) p -> a
 fromData = to . unData
 
diff --git a/src/Generic/Data/Internal/Generically.hs b/src/Generic/Data/Internal/Generically.hs
--- a/src/Generic/Data/Internal/Generically.hs
+++ b/src/Generic/Data/Internal/Generically.hs
@@ -36,9 +36,12 @@
 instance (Generic a, Semigroup (Rep a ())) => Semigroup (Generically a) where
   (<>) = gmappend
 
-instance (Generic a, Monoid (Rep a ())) => Monoid (Generically a) where
+-- | This uses the 'Semigroup' instance of the wrapped type 'a' to define 'mappend'.
+-- The purpose of this instance is to derive 'mempty', while remaining consistent
+-- with possibly custom 'Semigroup' instances.
+instance (Semigroup a, Generic a, Monoid (Rep a ())) => Monoid (Generically a) where
   mempty = gmempty
-  mappend = gmappend'
+  mappend (Generically x) (Generically y) = Generically (x <> y)
 
 instance (Generic a, GEnum StandardEnum (Rep a)) => Enum (Generically a) where
   toEnum = gtoEnum
diff --git a/src/Generic/Data/Internal/Microsurgery.hs b/src/Generic/Data/Internal/Microsurgery.hs
--- a/src/Generic/Data/Internal/Microsurgery.hs
+++ b/src/Generic/Data/Internal/Microsurgery.hs
@@ -42,7 +42,7 @@
 -- @
 type Surgery (s :: *) (a :: *) = Generically (Surgery' s a)
 
--- | See 'Surgery''.
+-- | See 'Surgery'.
 newtype Surgery' (s :: *) (a :: *) = Surgery' { unSurgery' :: a }
 
 instance (Generic a, Coercible (GSurgery s (Rep a)) (Rep a)) => Generic (Surgery' s a) where
@@ -78,7 +78,7 @@
 -- Concretely, set the last field of 'MetaCons' to 'False' and forget field
 -- names.
 --
--- This is a defunctionalized symbol, to be applied using 'GSurgery'.
+-- This is a defunctionalized symbol, applied using 'GSurgery' or 'Surgery'.
 data Derecordify :: *
 type instance GSurgery Derecordify f = GDerecordify f
 
@@ -112,7 +112,7 @@
 -- >
 -- > data Foo = Bar Baz
 --
--- This is a defunctionalized symbol, to be applied using 'GSurgery'.
+-- This is a defunctionalized symbol, applied using 'GSurgery' or 'Surgery'.
 data Typeage :: *
 type instance GSurgery Typeage (M1 D ('MetaData nm md pk _nt) f) = M1 D ('MetaData nm md pk 'False) f
 
@@ -150,7 +150,7 @@
 -- >
 -- > data Foo = Bar { bag :: Zap }
 --
--- This is a defunctionalized symbol, to be applied using 'GSurgery'.
+-- This is a defunctionalized symbol, applied using 'GSurgery' or 'Surgery'.
 data RenameFields (rnm :: *) :: *
 type instance GSurgery (RenameFields rnm) f = GRenameFields rnm f
 
@@ -171,7 +171,7 @@
 -- >
 -- > data Foo = Car { baz :: Zap }
 --
--- This is a defunctionalized symbol, to be applied using 'GSurgery'.
+-- This is a defunctionalized symbol, applied using 'GSurgery' or 'Surgery'.
 data RenameConstrs (rnm :: *) :: *
 type instance GSurgery (RenameConstrs rnm) f = GRenameConstrs rnm f
 
@@ -252,7 +252,7 @@
 -- | Apply a type constructor @f@ to every field type of a generic
 -- representation @r@.
 --
--- This is a defunctionalized symbol, to be applied using 'GSurgery'.
+-- This is a defunctionalized symbol, applied using 'GSurgery' or 'Surgery'.
 data OnFields (f :: * -> *) :: *
 type instance GSurgery (OnFields f) g = GOnFields f g
 
diff --git a/src/Generic/Data/Internal/Utils.hs b/src/Generic/Data/Internal/Utils.hs
--- a/src/Generic/Data/Internal/Utils.hs
+++ b/src/Generic/Data/Internal/Utils.hs
@@ -1,5 +1,6 @@
 {-# LANGUAGE
     BangPatterns,
+    FlexibleContexts,
     PolyKinds #-}
 
 module Generic.Data.Internal.Utils where
@@ -7,9 +8,25 @@
 import Data.Coerce
 import GHC.Generics
 
+-- | Convert between types with representationally equivalent generic
+-- representations.
+gcoerce
+  :: (Generic a, Generic b, Coercible (Rep a) (Rep b))
+  => a -> b
+gcoerce = to . coerce1 . from
+
+-- | Compose 'gcoerce' with a binary operation.
+gcoerceBinop
+  :: (Generic a, Generic b, Coercible (Rep a) (Rep b))
+  => (a -> a -> a) -> (b -> b -> b)
+gcoerceBinop f x y = gcoerce (f (gcoerce x) (gcoerce y))
+
 -- | Coerce while preserving the type index.
 coerce' :: Coercible (f x) (g x) => f x -> g x
 coerce' = coerce
+
+coerce1 :: Coercible f g => f x -> g x
+coerce1 = coerce
 
 -- | Elimination of @V1@.
 absurd1 :: V1 x -> a
diff --git a/test/unit.hs b/test/unit.hs
--- a/test/unit.hs
+++ b/test/unit.hs
@@ -16,6 +16,9 @@
 data P a = P a a
   deriving (Generic, Generic1)
 
+instance Semigroup a => Semigroup (P a) where
+  x <> y = unGenerically (Generically x <> Generically y)
+
 type PTy a = a -> a -> Generically (P a)
 
 p :: PTy a