Definition:Functor Evaluation Bifunctor

Definition
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 $G(f) \circ \eta_a = \eta_b \circ F(f)$

Also see

 * Functor Evaluation Bifunctor is Functor