---
type: "concept"
title: "Covariant hom functor"
slug: "covariant-hom-functor"
language: "en"
translationGroupId: "cms07238d0009ll01vbx7qt01"
domain: "Algebra"
status: "stub"
aliases: []
lastEditedBy: "ancient-tree"
---

Let $\mathcal{C}$ be a [[Locally small category|locally small category]]. For each object $A$ in $\mathcal{C}$, there is a [[Functor|functor]]

$$\operatorname{Hom}_{\mathcal{C}}(A,-):\mathcal{C}\rightarrow \mathbf{Set}$$

defined by:
- On objects: each object $X$ is sent to the set $\operatorname{Hom}_{\mathcal{C}}(A,X)$.
- On morphisms: a morphism $f:X\rightarrow Y$ is sent to the function
$$f_*=\operatorname{Hom}_{\mathcal{C}}(A, f): \operatorname{Hom}_{\mathcal{C}}(A, X) \rightarrow \operatorname{Hom}_{\mathcal{C}}(A, Y)$$
given by [[Post-composition|post-composition]]: $g\mapsto f\circ g$.