Functor

Hom-functor

The Hom-functor Hom𝖢 :𝖢𝐨𝐩 ×𝖢 →𝖲𝖾𝗍 is a bifunctor, contravariant in its first argument and covariant in its second, for a locally small category 𝖢. Often we use 𝖢 as a shorthand for Hom𝖢

On objects, it maps (𝐶,𝐶′) ∈Ob⁡(𝖢𝐨𝐩 ×𝖢) to the Hom-set of morphisms with domain 𝐶 and codomain 𝐶′.

The morphism map for fixed domain 𝐶 ∈𝖢0 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.

#state/tidy| #lang/en | #SemBr