Corollary 7.3.6.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories equipped with diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$, and suppose that $F_0$ admits a left Kan extension along $\delta $. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor and let $\beta : F_0 \rightarrow F \circ \delta $ be a natural transformation. The following conditions are equivalent:
- $(1)$
The natural transformation $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $.
- $(2)$
For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, G ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( F \circ \delta , G \circ \delta ) \xrightarrow { \circ [\beta ] } \operatorname{Hom}_{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}( F_0, G \circ \delta ) \]
is a homotopy equivalence of Kan complexes.
- $(3)$
For every functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F, G ) \rightarrow \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(K, \operatorname{\mathcal{D}})} }( F \circ \delta , G \circ \delta )) \xrightarrow { \circ [\beta ] } \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}}( F_0, G \circ \delta ) \]
is a bijection of sets.
Proof.
The implication $(1) \Rightarrow (2)$ follows from Proposition 7.3.6.1 and the implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3) \Rightarrow (1)$. By assumption, there exists a functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and a natural transformation $\beta ': F_0 \rightarrow F' \circ \delta $ which exhibits $F'$ as a left Kan extension of $F$ along $\delta $. Applying Proposition 7.3.6.1, we deduce that there exists a natural transformation $\gamma : F' \rightarrow F$ for which $\beta $ is a composition of $\beta '$ with the induced transformation $\gamma |_{K}: (F' \circ \delta ) \rightarrow (F \circ \delta )$. For each object $G \in \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F, G ) \ar [rr]^{ \circ [\gamma ] } \ar [dr] & & \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})} }( F', G ) \ar [dl] \\ & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Fun}(K,\operatorname{\mathcal{D}})}}( F_0, G \circ \delta ), & } \]
where the right vertical map is bijective. If condition $(3)$ is satisfied, then the left vertical map is also bijective. Allowing the functor $G$ to vary, it follows that the homotopy class $[\gamma ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }$, so that $\gamma $ is an isomorphism in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Invoking Remark 7.3.1.12, we conclude that $\beta $ exhibits $F$ as a left Kan extension of $F_0$ along $\delta $.
$\square$