Proposition 7.3.8.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be another functor of $\infty $-categories. Assume that $F = \overline{F}|_{\operatorname{\mathcal{C}}}$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Then $\overline{F}$ is a $U$-colimit diagram if and only if the composite map
\[ (\operatorname{\mathcal{C}}^{0})^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright } \xrightarrow { \overline{F} } \operatorname{\mathcal{D}} \]
is a $U$-colimit diagram.
Proof.
For each object $D \in \operatorname{\mathcal{D}}$, let $\underline{D} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{D}})$ denote the constant functor taking the value $D$. By virtue of Proposition 7.1.5.12, the functor $\overline{F}$ is a $U$-colimit diagram if and only if, for each $D \in \operatorname{\mathcal{D}}$, the upper half of the diagram
7.33
\begin{equation} \begin{gathered}\label{equation:Kan-extension-relative-colimit} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{D}})}( \overline{F}, \underline{D} ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{E}})}( U \circ \overline{F}, U \circ \underline{D} ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) }( F, \underline{D}|_{ \operatorname{\mathcal{C}}} ) \ar [d] \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( U \circ F, U \circ \underline{D}|_{\operatorname{\mathcal{C}}}) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}})}( F|_{ \operatorname{\mathcal{C}}^{0} }, \underline{D}|_{ \operatorname{\mathcal{C}}^{0} } ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^0, \operatorname{\mathcal{E}})}( U \circ F|_{ \operatorname{\mathcal{C}}^{0} }, U \circ \underline{D}|_{ \operatorname{\mathcal{C}}^{0} } )} \end{gathered} \end{equation}
is a homotopy pullback square. Since $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, Proposition 7.3.6.7 shows that the right half of the diagram is a homotopy pullback square. It follows that $\overline{F}$ is a $U$-colimit diagram if and only if the outer rectangle of (7.33) is a homotopy pullback square for each $D \in \operatorname{\mathcal{D}}$ (Proposition 3.4.1.11).
Let $v$ denote the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$. Let $\operatorname{\mathcal{C}}^{1}$ denote the cone $(\operatorname{\mathcal{C}}^0)^{\triangleright }$, which we regard as a full subcategory of $\operatorname{\mathcal{C}}^{\triangleright }$. Note that the functors $\underline{D}$, $\underline{D}|_{ \operatorname{\mathcal{C}}^{1}}$, $U \circ \underline{D}$ and $U \circ \underline{D}|_{ \operatorname{\mathcal{C}}^{1} }$ are right Kan extended from the cone point, so Corollary 7.3.6.9 implies that the restriction maps
\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{D}})}( \overline{F}, \underline{D} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{1}, \operatorname{\mathcal{D}})}( \overline{F}|_{ \operatorname{\mathcal{C}}^{1}}, \underline{D}|_{ \operatorname{\mathcal{C}}^{1}} ) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( \overline{F}(v), D ) \]
\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{E}})}( U \circ \overline{F}, U \circ \underline{D} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{1}, \operatorname{\mathcal{E}})}( U \circ \overline{F}|_{ \operatorname{\mathcal{C}}^{1}}, U \circ \underline{D}|_{ \operatorname{\mathcal{C}}^{1}} ) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{E}}}( (U \circ \overline{F}(v))), U(D) ) \]
are homotopy equivalences. It follows that the restriction map from the outer rectangle of (7.33) to the diagram
7.34
\begin{equation} \begin{gathered}\label{equation:Kan-extension-relative-colimit2} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^1, \operatorname{\mathcal{D}})}( \overline{F}|_{\operatorname{\mathcal{C}}^1} , \underline{D}|_{\operatorname{\mathcal{C}}^{1}} ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{D}}) }( F|_{\operatorname{\mathcal{C}}^{0}}, \underline{D}|_{ \operatorname{\mathcal{C}}^0} ) \ar [d] \\ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^1, \operatorname{\mathcal{E}})}( U \circ \overline{F}|_{\operatorname{\mathcal{C}}^1}, U \circ \underline{D}|_{\operatorname{\mathcal{C}}^{1}} ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^0, \operatorname{\mathcal{E}}) }( U \circ F, U \circ \underline{D}|_{\operatorname{\mathcal{C}}^0}) } \end{gathered} \end{equation}
is a levelwise homotopy equivalence. In particular, the outer rectangle of (7.33) is a homotopy pullback square if and only if (7.34) is a homotopy pullback square (Corollary 3.4.1.12). By virtue of Proposition 7.1.5.12, this is satisfied for every object $D \in \operatorname{\mathcal{D}}$ if and only if $F^{1}$ is a $U$-colimit diagram.
$\square$