Contravariant Hom Functor maps Colimits to Limits

Theorem

Let $\mathbf{Set}$ be the category of sets.

Let $\mathbf C$ be a locally small category.

Let $C$ be an object of $\mathbf C$.

Let $\map \hom {\cdot, C}: \mathbf C \to \mathbf{Set}$ be the contravariant hom functor based at $C$.

Then $\map \hom {\cdot, C}$ maps every colimit to a limit, in that:

$\map \hom { {\varinjlim \,}_j \,D_j, C} \cong {\varprojlim \,}_j \, \map \hom {D_j, C}$

for every diagram $D: \mathbf J \to \mathbf C$.

Proof

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Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.5$: Corollary $5.27$