nLab
hom-functor

For any locally small category $C$ and any object $c\in C$, the covariant hom-functor

$$\mathrm{hom}(c,-):C\to \mathrm{Set}$$hom(c,-) : C \to Set

is the functor from $C$ to Set sending any object $x\in C$ to the hom-set $\mathrm{hom}(c,x)$.

To fully describe this functor we must also say what it does to morphisms in $C$: $f:x\to y$ is sent to the morphism $f_{*}:\mathrm{hom}(c,x)\to \mathrm{hom}(c,y)$, which maps $\phi$ to $f\circ \phi$.

There is also a contravariant hom-functor

$$\mathrm{hom}(-,c):C^{\mathrm{op}}\to \mathrm{Set},$$hom(-,c) : C^{op} \to Set,

where $C^{\mathrm{op}}$ is the opposite category to $C$, which sends any object $x\in C^{\mathrm{op}}$ to the hom-set $\mathrm{hom}(x,c)$.

And indeed, both the hom-functors just mentioned can easily be obtained from the more general functor, also called a hom-functor:

$$\mathrm{hom}:C^{\mathrm{op}}\times C\to \mathrm{Set},$$hom : C^{op} \times C \to Set,

where $C^{\mathrm{op}}\times C$ is the product category? of $C^{\mathrm{op}}$ and $C$, which sends any object $(x,y)\in C^{\mathrm{op}}\times C$ to the hom-set $\mathrm{hom}(x,y)$.