Kerodon

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Remark 7.3.6.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty$-categories equipped with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$. It follows from Corollary 7.3.6.5 that if $F_0$ admits a left Kan extension $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ along $\delta$, then the isomorphism class of the functor $F$ is uniquely determined: it is characterized by the requirement that it corepresents the functor

$\mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}},\operatorname{\mathcal{D}})} \rightarrow \operatorname{Set}\quad \quad G \mapsto \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}}( F_0, G \circ \delta ).$