Replacing a morphism with composition with this morphism

I have a certain category. I feel it is better to study the functor $x\mapsto f\circ x$ (where $\circ$ is the composition in my category) than the morphism $f$ itself.

1. How is it called when a morphism $f$ is replaced with the functor $x\mapsto f\circ x$? Is it Yoneda embedding?

2. Should I take only endomorphisms on the source of $f$ for $x$ (or all morphisms whose destination is equal to the source of $f$)?

But isn't it a partial functor? It takes only these $x$ that $\operatorname{Dst} x=\operatorname{Src} f$ (or even only some endomorphisms if we restrict). Is there are name for such partial functors? (I've never heard about partial functors, but now I feel this topic is important.)

-
I've realized that if it is defined only for endomorphisms on the source of $f$, it can nevertheless be continued (in a definite way) for all morphisms whose destination is equal to the source of $f$ (by composing with the identity morphism). So these are bijective, and I think I should introduce a notation for both. – porton Sep 21 '13 at 15:57

This is mostly called the hom functor.

For each fixed object $A$, we have $\hom(A,-)$ which takes the morphism $f$ to the map $x\mapsto f\circ x$ (for morphisms $x:A\to\mathrm{Src}\,f$).

If we let $A$ vary, we arrive to the full hom functor $\hom(-,-):\mathcal C^{op}\times\mathcal C\to\mathcal Set$.

If $f:U\to V$ is fixed, it yields to a natural transformation $\hom(-,U) \,\to\,\hom(-,V)$.

If $A$ is also fixed therein, it is just a function $\hom(A,U)\to\hom(A,V)$.

Well, the Yoneda embedding is indeed something like this, it maps an object $X$ to the functor $\hom(-,X)$ and a morphism $U\to V$ to a natural transformation $\ \hom(-,U)\to\hom(-,V)$.

-
In fact, we might even call that natural transformation $\hom(-,f)$. – Hurkyl Sep 23 '13 at 11:17