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Corollary 11.5.0.42. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(\mathscr {H}, \alpha )$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$, let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. Then the following conditions are equivalent:

$(1)$

The morphism $f$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$

$(2)$

The vertex $\alpha (f) \in \mathscr {H}(X,Y)$ exhibits the functor $\mathscr {H}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by $Y$.

$(3)$

The vertex $\alpha (f) \in \mathscr {H}(X,Y)$ exhibits the functor $\mathscr {H}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by $X$.