Corollary 7.1.6.7 (02XG)

Corollary 7.1.6.7. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories, let $B$ and $K$ be simplicial sets, and suppose we are given a lifting problem

7.5

\begin{equation} \begin{gathered}\label{equation:relative-colimit-pointwise-existence} \xymatrix@R =50pt@C=50pt{ B \times K \ar [r]^-{ f } \ar [d] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ B \times K^{\triangleright } \ar [r]^-{\overline{g}} \ar@ {-->}[ur]^{ \overline{f} } & \operatorname{\mathcal{D}}} \end{gathered} \end{equation}

Assume that, for each vertex $b \in B$, the restriction $f|_{\{ b\} \times K}$ can be extended to a $U$-colimit diagram $\overline{f}_{b}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $U \circ \overline{f}_{b} = \overline{g}|_{ \{ b\} \times K^{\triangleright } }$. Then the lifting problem (7.5) admits a solution $\overline{f}: B \times K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}|_{ \{ b\} \times K^{\triangleright } } = \overline{f}_{b}$ for each $b \in B$.

Proof. Apply Corollary 7.1.6.6 in the special case where $A = \operatorname{sk}_0(B)$ is the $0$-skeleton of $B$. $\square$