Suppose we are in the category $\mathcal{C}$ of (finite dimensional if necessary) vector spaces over a fixed field. My question then is, is the map $F$ which takes a vector space $V$ and returns it's endomorphism ring $\text{End}(V)=\text{Hom}(V,V)$ a functor?
For $F$ to be a functor, it needs to act on objects and morphisms in a compatible manner. It is clear what it sends each object to, but I can't seem to figure out what it might send each morphism to.
Consider a morphism $T: V \to W$. Then we need $F(T):\text{End}(V) \to \text{End}(W)$. If we were able to ensure that $T$ is invertible, there seem to be the natural choice of: $$ F(T)(\phi) = T^{-1} \circ \phi \circ T$$
But as it stands, $T$ need not be invertible. I am getting the feeling that maybe there is no such natural choice which makes $F$ a functor, in which case, I am wondering, is that an actual theorem?