Class that represents various endomorphism representation. In other words anything that encodes (a -> a) can be instance of this class.

Here are some important instances with not so obvious definitions.

instance AnEndo (Proxy a) where
    type EndoOperatesOn (Proxy a) = a

    anEndo    _ = mempty -- = Endo id
    aDualEndo _ = mempty

It got quite common to use Proxy data type as an explicit way to pass types around. Above instance allows you to restrict type of result of endomorphism folding, to some extent.

instance AnEndo a => AnEndo (Maybe a) where
    type EndoOperatesOn (Maybe a) = EndoOperatesOn a

    anEndo Nothing  = mempty -- = Endo id
    anEndo (Just e) = anEndo e

    -- Definition of aDualEndo is analogous.

Instance for Maybe lets us conditionally inject endomorphism in to a folding chain.

instance AnEndo a => AnEndo (Identity a) where
    type EndoOperatesOn (Identity a) = EndoOperatesOn a

    anEndo (Identity e) = anEndo e
    aDualEndo (Identity e) = aDualEndo e

Above instance allows us to discard Identity wrapper, which is commonly used in data types that are parametrized by functor or monad.

Wrapper for Foldable types. Used to provide instances that work for all Foldable types without the need for OverlappingInstances language extension.

Use Endo (possibly result of foldEndo) and use it to create value of different type.

Examples:

embedEndoWith tell
    :: (Monad m, AnEndo e, w ~ EndoOperatesOn e)
    => e
    -> WriterT (Endo w) m ()

embedEndoWith (modify . appEndo)
    :: (Monad m, AnEndo e, s ~ EndoOperatesOn e)
    => e
    -> StateT s m ()

See also embedDualEndoWith.

Dual to embedEndoWith, which uses aDualEndo instead of anEndo.