Proposition 7.1.6.24. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $D \in \operatorname{\mathcal{D}}$ be an object, and let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ be a diagram. Then $\overline{f}$ is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it satisfies the following condition:
- $(\ast )$
For every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the morphism
\[ K^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}}_{D} \hookrightarrow \Delta ^{1} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}} \]
is a $U'$-colimit diagram, where $U': \Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1$ is given by projection onto the first factor.
Proof.
Assume that condition $(\ast )$ is satisfied; we will show that $\overline{f}$ is a $U$-colimit diagram (the converse follows from Proposition 7.1.6.21). Set $f = \overline{f}|_{K}$. By virtue of Proposition 7.1.6.12, it will suffice to show that for each vertex $C \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes
7.4
\begin{equation} \begin{gathered}\label{equation:relative-colimit-by-fiber-preliminary} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) }( \overline{f}, \underline{C}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f, \underline{C}|_{K} ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ \overline{f}, U \circ \underline{C} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C}|_{K}) } \end{gathered} \end{equation}
is a homotopy pullback square, where $\underline{C} \in \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$ is the constant diagram taking the value $C$. Since $U$ is an inner fibration, the vertical maps in (7.4) are Kan fibrations (Proposition 4.6.1.21 and Corollary 4.1.4.3). Using the criterion of Example 3.4.1.4, it will suffice to show that for every vertex $u \in \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ \overline{f}, U \circ \underline{C} )$, the induced map
\[ \xymatrix@R =50pt@C=50pt{ \{ u\} \times _{\operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ \overline{f}, U \circ \underline{C} )} \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( \overline{f}, \underline{C}) \ar [d]^{\theta _ u} \\ \{ u\} \times _{ \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C}|_{K}) } \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f, \underline{C}|_{K} )} \]
is a homotopy equivalence of Kan complexes. Set $D' = U(C)$, so that $u$ can be identified with a morphism of simplicial sets $K^{\triangleright } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, D' )$, and the condition that $\theta _ u$ is a homotopy equivalence depends only on the homotopy class of $u$. Since the simplicial set $K^{\triangleright }$ is weakly contractible (Example 4.3.7.11), we may assume without loss of generality that $u: K^{\triangleright } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, D' )$ is the constant map taking the value $e$, for some morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$. The desired result now follows from $(\ast )$.
$\square$