Corollary 8.4.5.11 (05ZZ)

Corollary 8.4.5.11. Let $\mathbb {K}$ be a collection of simplicial sets, let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{D}}$ is $\mathbb {K}$-cocomplete. Suppose we are given a functor $F: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\alpha : f \rightarrow F \circ h$, as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \widehat{\operatorname{\mathcal{C}}} \ar [dr]^{F} \ar@ {<=}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha }_{\sim } & \\ \operatorname{\mathcal{C}}\ar [ur]^{h} \ar [rr]_{f} & & \operatorname{\mathcal{D}}. } \]

The following conditions are equivalent:

$(1)$: The natural transformation $\alpha $ exists $F$ as a $\mathbb {K}$-cocontinuous extension of $f$ (see Remark 8.4.5.5). That is, $F$ is $\mathbb {K}$-cocontinuous and $\alpha $ is an isomorphism.
$(2)$: The natural transformation $\alpha $ exhibits $F$ as a left Kan extension of $f$ along $h$.

Proof. By virtue of Proposition 8.4.5.8, we may assume without loss of generality that the functor $h$ is obtained from Construction 8.4.5.6. In this case, the equivalence of $(1)$ and $(2)$ is a reformulation of Lemma 8.4.5.10. $\square$