\begin{definition}[Definition:Hom Bifunctor]
{{tidy|Use standardised naming of category and elements}}
Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
The '''hom bifunctor''' on $\mathbf C$ is the covariant functor $\map {\operatorname {Hom} } {-, -} : \mathbf C^{\operatorname {op} } \times \mathbf C \to \mathbf {Set}$ from the product with the opposite category to the category of sets such that:
:$(1): \quad \map {\operatorname {Hom} } {a, b}$ is the hom class
:$(2): \quad$ If $\tuple {f^{\operatorname {op} }, g}: \tuple {a, b} \to \tuple {c, d}$ is a morphism, $\map {\operatorname {Hom} } {f^{\operatorname{op} }, g}: \map {\operatorname {Hom} } {a, b} \to \map {\operatorname {Hom} } {c, d}$ is $f_* \circ g^*$, the postcomposition with $g$ composed with the precomposition with $f$.
\end{definition}