Hom-Functor

Let $\mathcal{C}$ be a locally-small category, The hom-functor $\mathrm{hom}:{\mathcal{C}}^{\mathrm{op}}\text{texttimes}\mathcal{C}\to \mathbf{Set}$ is defined as:

$$\begin{array}{r}\mathrm{hom}(a,b)=\mathcal{C}[a,b]\\ \mathrm{hom}(f,g)=h\end{array}$$ $$\begin{array}{rl}\text{Where }& h:\mathcal{C}[b,c]\to \mathcal{C}[a,d]\\ & h:j\mapsto g\circ j\circ f\\ \text{For }& f:a\to b\\ & g:c\to d\end{array}$$

Fixing an object $a:\mathcal{C}$, we can produce two hom-functors from partially applying $\mathrm{hom}$ to the left and right arguments:

Covariant Hom-Functor

Have $\mathrm{hom}(a,-):\mathcal{C}\to \mathbf{Set}$.

$$\begin{array}{r}hom(a,b)=\mathcal{C}[a,b]\\ hom(a,f):\lambda g:a\to b.f\circ g\\ \text{where }f:b\to c\end{array}$$

Contravariant Hom-Functor

Have $\mathrm{hom}(-,a):\mathcal{C}\to \mathbf{Set}$.

$$\begin{array}{r}hom(b,a)=\mathcal{C}[b,a]\\ hom(f,a):\lambda g:c\to a.g\circ f\\ \text{where }f:b\to c\end{array}$$

Resources

https://en.wikipedia.org/wiki/Hom_functor https://ncatlab.org/nlab/show/hom-functor