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Proposition 7.3.7.18. Let $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}_1$, $\operatorname{\mathcal{C}}_2$, and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Suppose we are given functors $F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}$ for $0 \leq i \leq 2$, functors $G: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_1$ and $H: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_2$, and natural transformations

\[ \alpha : F_0 \rightarrow F_1 \circ G \quad \quad \beta : F_1 \rightarrow F_2 \circ H, \]

where $\alpha $ exhibits $F_1$ as a left Kan extension of $F_0$ along $G$. The following conditions are equivalent:

$(1)$

The natural transformation $\beta $ exhibits $F_2$ as a left Kan extension of $F_1$ along $H$.

$(2)$

Let $\gamma : F_0 \rightarrow F_2 \circ H \circ G$ be a composition of $\alpha $ with $\beta |_{\operatorname{\mathcal{C}}^{0}}$ (formed in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}})$). Then $\gamma $ exhibits $F_2$ as a left Kan extension of $F_0$ along $H \circ G$.

Proof. Let $\operatorname{\mathcal{C}}$ denote the iterated relative join $(\operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1) \star _{\operatorname{\mathcal{C}}_2} \operatorname{\mathcal{C}}_2$, so that we have a cocartesian fibration of $\infty $-categories $\pi : \operatorname{\mathcal{C}}\rightarrow \Delta ^2$ having fibers $\pi ^{-1} \{ i\} = \operatorname{\mathcal{C}}_ i$ for $0 \leq i \leq 2$ (see Lemma 5.2.3.17). For $0 \leq i < j \leq 2$, let $\operatorname{\mathcal{C}}_{ij}$ denote the fiber product $\operatorname{N}_{\bullet }( \{ i < j \} ) \times _{\Delta ^2} \operatorname{\mathcal{C}}$, which we will identify with $\operatorname{\mathcal{C}}_{i} \star _{\operatorname{\mathcal{C}}_ j} \operatorname{\mathcal{C}}_ j$. By virtue of Remark 7.3.1.9, we are free to replace $\alpha $ and $\beta $ by homotopic natural transformations. We can therefore assume that there exist functors

\[ F_{01}: \operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{D}}\quad \quad F_{12}: \operatorname{\mathcal{C}}_{12} \rightarrow \operatorname{\mathcal{D}} \]

satisfying $F_{01}|_{\operatorname{\mathcal{C}}_0} = F_0$, $F_{01}|_{\operatorname{\mathcal{C}}_1} = F_1 = F_{12}|_{\operatorname{\mathcal{C}}_1}$, and $F_{12}|_{\operatorname{\mathcal{C}}_2} = F_2$, where $\alpha $ and $\beta $ are given by the composite maps

\[ \Delta ^1 \times \operatorname{\mathcal{C}}_0 \simeq \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1 \xrightarrow {F_{01}} \operatorname{\mathcal{D}} \]

\[ \Delta ^1 \times \operatorname{\mathcal{C}}_1 \simeq \operatorname{\mathcal{C}}_1 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_1 \star _{\operatorname{\mathcal{C}}_2} \operatorname{\mathcal{C}}_2 \xrightarrow {F_{12}} \operatorname{\mathcal{D}} \]

(see Warning 7.3.2.11). Note that $F_{01}$ and $F_{12}$ can be amalgamated to a morphism of simplicial sets $F': \Lambda ^{2}_{1} \times _{\Delta ^1} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Since $\pi $ is a cocartesian fibration, the inclusion map $\Lambda ^{2}_{1} \times _{\Delta ^1} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}$ is a categorical equivalence (Proposition 5.3.6.1). Applying Lemma 4.5.5.2, we can extend $F'$ to a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

Let $F_{02}$ denote the restriction of $F$ to $\operatorname{\mathcal{C}}_{02}$, and let $\gamma : F_0 \rightarrow F_2 \circ H \circ G$ denote the natural transformation given by the composite map

\[ \Delta ^1 \times \operatorname{\mathcal{C}}_0 \simeq \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_2} \operatorname{\mathcal{C}}_2 \xrightarrow {F_{02}} \operatorname{\mathcal{D}}. \]

Note that the composite map

\[ \Delta ^2 \times \operatorname{\mathcal{C}}_0 \simeq (\operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0) \star _{\operatorname{\mathcal{C}}_0} \operatorname{\mathcal{C}}_0 \rightarrow (\operatorname{\mathcal{C}}_0 \star _{\operatorname{\mathcal{C}}_1} \operatorname{\mathcal{C}}_1) \star _{\operatorname{\mathcal{C}}_2} \operatorname{\mathcal{C}}_2 \xrightarrow {F} \operatorname{\mathcal{D}} \]

can be regarded as a $2$-simplex of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_0, \operatorname{\mathcal{D}})$, which witnesses $\gamma $ as a composition of $\alpha $ with $\beta |_{\operatorname{\mathcal{C}}_0}$. Applying Proposition 7.3.2.10, we see that $(1)$ and $(2)$ can be reformulated as follows:

$(1')$

The functor $F_{12}: \operatorname{\mathcal{C}}_{12} \rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_1$.

$(2')$

The functor $F_{02}: \operatorname{\mathcal{C}}_{02} \rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_0$.

By assumption, the natural transformation $\alpha $ exhibits $F_1$ as a left Kan extension of $F_0$ along $G$. Applying Proposition 7.3.2.10, we see that the functor $F_{01}$ is left Kan extended from $\operatorname{\mathcal{C}}_0$. In particular, $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ at every object of the full subcategory $\operatorname{\mathcal{C}}_1 \subseteq \operatorname{\mathcal{C}}$. It follows that $(2')$ is equivalent to the following:

$(2'')$

The functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_0$.

Using Corollary 7.3.7.8, we see that $(2'')$ is equivalent to the following:

$(1'')$

The functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_{01}$.

To complete the proof, it will suffice to show that conditions $(1')$ and $(1'')$ are equivalent. We will prove something slightly more precise: for every object $X \in \operatorname{\mathcal{C}}_2$, the conditions are equivalent:

$(1'_ X)$

The functor $F_{12}: \operatorname{\mathcal{C}}_{12} \rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_1$ at $X$.

$(1''_{X})$

The functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left Kan extended from $\operatorname{\mathcal{C}}_{01}$ at $X$.

Let us regard the object $X$ as fixed, and let $F_{X}$ denote the composite map

\[ (\operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X})^{\triangleright } \hookrightarrow ( \operatorname{\mathcal{C}}_{/X} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}. \]

We wish to show that $F_{X}$ is a colimit diagram in $\operatorname{\mathcal{D}}$ if and only if its restriction to $( \operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} )^{\triangleright }$ is a colimit diagram in $\operatorname{\mathcal{D}}$. By virtue of Corollary 7.2.2.3, it will suffice to show that the inclusion map $\operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} \hookrightarrow \operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X}$ is right cofinal. This follows by applying Proposition 7.2.3.13 to the upper square of the pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} \ar [r] \ar [d] & \{ 1\} \ar [d] \\ \operatorname{\mathcal{C}}_{01} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/X} \ar [r] \ar [d] & \Delta ^1 \ar [d] \\ \operatorname{\mathcal{C}}_{/X} \ar [r]^-{\pi '} & \Delta ^2, } \]

where $\pi '$ denotes the composite map $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}\rightarrow \Delta ^2$ (which is a cocartesian fibration by virtue of Proposition 5.1.4.19). $\square$