# Kerodon

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Remark 8.3.1.5. Let $\operatorname{\mathcal{C}}$ be a small $\infty$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. In §5.7.6, we showed that there exists a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresented by $X$, and that $\mathscr {F}$ is uniquely determined up to isomorphism (Theorem 5.7.6.13). Proposition 8.3.1.3 can be regarded as a more refined version of this uniqueness assertion: the functor $\mathscr {F}$ is characterized, up to isomorphism, by the requirement that it corepresents the evaluation functor

$\operatorname{ev}_{X}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad \mathscr {G} \mapsto \mathscr {G}(X).$