Covariant hom functor
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7/25/2026, 10:07:03 AM · Ancient Tree
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Let $\mathcal{C}$ be a [[Locally small category|locally small category]]. For each object $A$ in $\mathcal{C}$, there is a [[Functor|functor]]2
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$$\operatorname{Hom}_{\mathcal{C}}(A,-):\mathcal{C}\rightarrow \mathbf{Set}$$4
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defined by:6
- On objects: each object $X$ is sent to the set $\operatorname{Hom}_{\mathcal{C}}(A,X)$.7
- On morphisms: a morphism $f:X\rightarrow Y$ is sent to the function8
$$f_*=\operatorname{Hom}_{\mathcal{C}}(A, f): \operatorname{Hom}_{\mathcal{C}}(A, X) \rightarrow \operatorname{Hom}_{\mathcal{C}}(A, Y)$$9
given by post-composition: $g\mapsto f\circ g$.9
given by [[Post-composition|post-composition]]: $g\mapsto f\circ g$.Revision 585
7/25/2026, 9:56:10 AM · Ancient Tree
Concept created
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: $g\mapsto f\circ g$.