Composition Functor

Definition

Let $C$ be a category. For a fixed object $Z \in C$ , the assignment

C [Z, -] : C \Rightarrow Set

is the covariant hom-functor. On a morphism $f : X \to Y$ , it acts by postcomposition:

C [Z, f] = f \circ - : C [Z, X] \to C [Z, Y],

sending a morphism $h : Z \to X$ to $f \circ h : Z \to Y$ .

Dually, for a fixed object $X \in C$ , the assignment

C [-, X] : C^{o p} \Rightarrow Set

is the contravariant hom-functor. A morphism $g : Z^{'} \to Z$ acts by precomposition:

- \circ g : C [Z, X] \to C [Z^{'}, X], h \mapsto h \circ g .