Kerodon

Remark 5.6.6.7 (Uniqueness of the Corepresenting Object). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories which is corepresentable by an object $X \in \operatorname{\mathcal{C}}$. Let $Y$ be another object of $\operatorname{\mathcal{C}}$. Then $\mathscr {F}$ is corepresentable by $Y$ if and only if $Y$ is isomorphic to $X$. The “if” direction follows immediately from Remark 5.6.6.6. Conversely, suppose that $\mathscr {F}$ is corepresentable by $Y$. Choose vertices $x \in \mathscr {F}(X)$ and $y \in \mathscr {F}(Y)$ which exhibit $\mathscr {F}$ as corepresented by $X$ and $Y$, respectively. Since evaluation at $x$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \mathscr {F}(Y)$, we can choose a morphism $u: X \rightarrow Y$ such that $\mathscr {F}(u)(x)$ and $y$ belong to the same connected component of $\mathscr {F}(Y)$. Then $\mathscr {F}(u)(x)$ also exhibits $\mathscr {F}$ as corepresented by $Y$ (Remark 5.6.6.3), so that $u$ is an isomorphism in $\operatorname{\mathcal{C}}$ by virtue of Remark 5.6.6.6.