\begin{definition}[Definition:Functor Evaluation Bifunctor]
Let $C$ and $D$ be categories.
Let $\operatorname{Funct}(C, D)$ be their covariant functor category.
Let $\operatorname{Funct}(C, D) \times C$ be the product category.
The '''evaluation bifunctor''' $\operatorname{ev} : \operatorname{Funct}(C, D) \times C \to D$ is the covariant functor that sends:
*an object $(F, a)$ to $F(a)$
*a morphism $(\eta, f) : (F, a) \to (G, b)$ to $G(f) \circ \eta_a = \eta_b \circ F(f)$
\end{definition}