Kerodon

Remark 11.5.0.42. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is essentially small and $\operatorname{\mathcal{D}}$ admits small colimits. Fix a covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. Theorem 8.4.0.3 implies that $f$ is isomorphic to the composition $F \circ h_{\bullet }$, for some functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{D}}$ which preserves small colimits. Moreover, the functor $F$ is uniquely determined up to isomorphism.