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Theorem 5.7.6.13. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category. Then, for every object $X \in \operatorname{\mathcal{C}}$, there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by $X$. Moreover, the functor $\mathscr {F}$ is uniquely determined up to isomorphism.

Proof of Theorem 5.7.6.13. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We wish to show that there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by $X$, and that $\mathscr {F}$ is uniquely determined up to isomorphism (as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$. By virtue of Proposition 5.7.6.21 and Corollary 5.7.0.6, this is equivalent to the assertion that there exists a left fibration $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ together with an initial object $\widetilde{X} \in \operatorname{\mathcal{D}}$ satisfying $U( \widetilde{X}) = X$, and that the left fibration $U$ is uniquely determined up to equivalence (in the sense of Definition 5.1.6.1). To prove existence, we can take $\operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}_{X/}$ and $\widetilde{X}$ to be the identity morphism $\operatorname{id}_{X}$ (Proposition 4.6.6.23). The uniqueness assertion follows from Proposition 5.7.6.19. $\square$