Is the identity functor the terminal object of the category of endofunctors on C?

Question

It seems to me not, since this would seem to imply that for all functors F and all objects A in C there exists a morphism F(A) -> A (making all functors co-pointed?). However, intuitively it seems like the identity functor acts like a terminal object; a monad M on C is a monoid on [C, C] where the "unit" is a natural transformation η : I -> M, while for a monoidal set S in Set the unit is a function e : 1 -> S. So am I misunderstanding something, or are my intuitions leading me astray?

Answer 1

The difference between those two examples is that in $\textbf{Set}$ the monoidal operation is the categorical product (so the identity object is the terminal object), whereas this is not true in the category of endofunctors on $\mathbf{C}$. (I believe the latter has a product if and only if $\mathbf{C}$ does, and then it is the pointwise product. It follows that the terminal object, if it exists, is the functor which sends all objects to $\mathbf{1}$ and all morphisms to the unique morphism $\mathbf{1} \to \mathbf{1}$. In particular, it's not the identity functor.)