Remark 5.7.6.6. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories, let $u: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, and let $x \in \mathscr {F}(X)$ be a vertex having image $y = \mathscr {F}(u)(x) \in \mathscr {F}(Y)$. Then any two of the following conditions imply the third:

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

The vertex $y$ exhibits the functor $\mathscr {F}$ as corepresented by $Y$.

The morphism $u$ is an isomorphism.