Are all Haskell functors endofunctors?

Question

I\'m a bit confused, and need someone to set me straight. Lets outline my current understanding:

Where E is an endofunctor, and A is some categ

Answer 1

A possibly relevant (or at least interesting) discussion specifically regarding monads is found in the paper "Monads need not be endofunctors":

http://www.cs.nott.ac.uk/~txa/publ/Relative_Monads.pdf