Covariant Hom Functor is Continuous
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Theorem
Let $\mathbf{Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $C$ be an object of $\mathbf C$, and let $\hom \paren {C, \cdot}: \mathbf C \to \mathbf{Set}$ be the covariant hom functor based at $C$.
Then $\hom \paren {C, \cdot}$ is a continuous functor.
Proof
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Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.5$: Proposition $5.25$