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In category theory, hom-sets, are sets of morphisms between objects. Given objects $A$ and $B$ in a locally small category, the hom-set $\text{Hom}(A,B)$ is the set of all morphisms $\{\rightarrow_0,$$\rightarrow_1,$$\rightarrow_2,$$\dots,$$\rightarrow_n\}$ from A to B.

$$AB$$

Hom-sets itself give rise to another category where hom-sets are objects and arrows between hom-sets are hom-set morphisms. A hom-set morphism is defined as:

$\text{Hom}(A,f) : \text{Hom}(A,B) \rightarrow \text{Hom}(A,D)$, $f'$ $\rightarrow$ $f \circ {f'}$ for $f : B \rightarrow D$

$\text{Hom}(h,B) : \text{Hom}(A,B) \rightarrow \text{Hom}(C,B)$, $h'$ $\rightarrow$ $h' \circ {h}$ for $h : C \rightarrow A$

such that the following diagram commutes:

$$\text{Hom}(A,B)\text{Hom}(h,f)\text{Hom}(A,f)\text{Hom}(h,B)\text{Hom}(C,B)Hom(C,f)\text{Hom}(A,D)\text{Hom}(h,D)\text{Hom}(C,D)$$

This can be translated to Hask, the category with Haskell types as objects and functions as morphisms. The previous morphisms $f,h$ are now functions:

f :: b -> d
h :: c -> a

and $\text{Hom}(A,B) = a \rightarrow b$, such that:

$$a\to bc\to a\to b\to da\to b\to dc\to a\to bc\to bc\to b\to da\to dc\to a\to dc\to d$$

Let’s have a closer look at the morphism from $a \rightarrow b$ to $c \rightarrow b$. So we have a $a \rightarrow b$ and a morphisms that tells us we can go $c \rightarrow a \rightarrow b$ resulting in $c \rightarrow b$. Now, this makes sense if we remember how function composition works:

$$c{f}^{\prime }{g}^{\prime }\circ f^{\prime }a{g}^{\prime }b$$

We can see that $c \rightarrow b$ is the function composition $g' \circ {f'}$. Thus, a hom-set morphism $\text{Hom}(f,-)$ is simply morphism pre-composition $f \circ -$ and $\text{Hom}(-,f)$ is just morphism post-composition $- \circ f$. Furthermore, $\text{Hom}(A,-)$, which you can think of as $A \rightarrow$, is called a covariant hom-functor or representable functor from the category $C$ to the category $Set$, $x \rightarrow \text{Hom}_C(A,x)$ with $x \in C$ and $\text{Hom}_C(A,x) \in Set$.

When the second position is fixed $\text{Hom}(-,B): C^{op} \rightarrow Set$ (which you can think of as: $\rightarrow B$) then it’s called a contravariant functor. Consequently, if non of the arguments of the hom-functor are fixed $\text{Hom}(−,−): C^{op} \times C \rightarrow Set$ then we have a hom bifunctor also called profunctor, see for instance the diagonal arrow $\text{Hom}(h,f)$ with pre- and post-composition.