Why Functor class has no return function?

Question

From categorical point of view, functor is pair of two maps (one between objects and another between arrows of categories), following some axioms.

I have assumed, wh

Answer 1

Objects of a category are not the same as objects in a OO programming language (we prefer to call those values in Haskell; what they mean in category theory was discussed here). Rather, the objects of Hask are types. Haskell Functors are endofunctors in Hask, i.e. associate types to types, by the following means:

Prelude> :k Maybe
Maybe :: * -> *
Prelude> :k Int
Int :: *
Prelude> :k Maybe Int
Maybe Int :: *

OTOH, the arrows of Hask are in fact values, of some function type a -> b. These are associated in the following way:

fmap :: ( Functor (f ::   t     ->     f t       {- type-level  -} ) )
             =>         (a->b)  ->  fmap(a->b)   {- value-level -}
                     ≡  (a->b)  ->  (f a->f b)