Hom-functor

The Hom-functor \cat C : \op{\cat C} \times \cat C \to \Set is a bifunctor, contravariant in its first argument and covariant in its second, for a locally small category $𝖢$ .

On objects, it maps $(𝐶, 𝐶') \in O b (𝖢 𝐨 𝐩 \times 𝖢)$ to the Hom-set of morphisms with domain $𝐶$ and codomain $𝐶'$ .

The morphism map for fixed domain $𝐶 \in O b 𝖢$ is the covariant pushforward

while the morphism map for fixed codomain is the contravariant pullback

Since the following diagram commutes

this indeed forms a bifunctor.