---
type: "concept"
title: "Contravariant hom functor"
slug: "contravariant-hom-functor"
language: "en"
translationGroupId: "cms07hj4q000dll01pk3vtg4b"
domain: "Algebra"
status: "stub"
aliases: []
lastEditedBy: "ancient-tree"
---

Let $\mathcal{C}$ be a [[Locally small category|locally small category]]. For each object $B$ in $\mathcal{C}$, there is a [[Functor|functor]]
$$\operatorname{Hom}_{\mathcal{C}}(-,B):\mathcal{C}^{\operatorname{op}}\rightarrow \mathbf{Set}$$
defined by:
- On objects: each object $X$ is sent to the set $\operatorname{Hom}_{\mathcal{C}}(X,B)$.
- On morphisms: a morphism $f:X\rightarrow Y$ is sent to the function
$$f^*=\operatorname{Hom}_{\mathcal{C}}(f, B): \operatorname{Hom}_{\mathcal{C}}(Y, B) \rightarrow \operatorname{Hom}_{\mathcal{C}}(X, B)$$
given by [[Pre-composition|pre-composition]]: $g\mapsto g\circ f$.