E is an endofunctor, and
A is some category:
E : A -> A.
Since all types and morphisms in Haskell are in the
Hask category, is not any functor in Haskell also an endofunctor?
F : Hask -> Hask.
You may want to clarify whether you're asking about "functors in Haskell", or
Functors. It's not always clear what category is being assumed when Category Theory terms are used in Haskell.
但是，是的，默认假设是Hask，它被视为Haskell类型的范畴，具有函数作为态射。在那种情况下，在Hask上的endofunctor F将任何类型A映射为F（A），将两个A和B类型之间的任何函数f映射为某些F（A）和F（B）之间的函数F（f）。 。
If we then limit ourselves to only those endofunctors which map any type
a to a type
(f a) where
f is a type constructor with kind
* -> *, then we can describe the associated map for functions as a higher-order function with type
(a -> b) -> (f a -> f b), which is of course the type class called
However, one can easily imagine well-behaved endofunctors on Hask which can't be written (directly) as an instance of
Functor, such as a functor mapping a type
Either a t. And while there's obviously not much sense in a functor from Hask to some other category entirely, it's reasonable to consider a (contravariant) functor from Hask to Haskop.
Beyond that, instances of
Functor necessarily map from the entire category Hask onto some subset of it that, thus, also forms a category. But it's also reasonable to talk about functors between subsets of Hask. For instance, consider a functor that sends types
Maybe a to
You may wish to peruse the
category-extras package, which provides some Category Theory-inspired structures embedded within Hask instead of assuming the entirety of it.