Kerodon

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Remark 5.7.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a morphism in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$, and let $x$ be a vertex of the Kan complex $\mathscr {F}(X)$. Then any two of the following conditions imply the third:

The vertex $x \in \mathscr {F}(X)$ exhibits the functor $\mathscr {F}$ as corepresented by $X$.

The vertex $\alpha (x) \in \mathscr {G}(X)$ exhibits the functor $\mathscr {G}$ as corepresented by $X$.

The natural transformation $\alpha $ is an isomorphism.

In particular, if $\mathscr {F}$ and $\mathscr {G}$ are isomorphic objects of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$, then $\mathscr {F}$ is corepresentable by $X$ if and only if $\mathscr {G}$ is corepresentable by $X$.