Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.2.2.14. Suppose we are given lifting problem

7.9
\begin{equation} \begin{gathered}\label{equation:easy-relative-colimit-existence} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ f } \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}^{\triangleleft } \ar [r] \ar@ {-->}[ur]^{\overline{f}} & \operatorname{\mathcal{E}}, } \end{gathered} \end{equation}

where $\operatorname{\mathcal{C}}$ is an $\infty $-category and $U$ is a cartesian fibration of $\infty $-categories. If $\operatorname{\mathcal{C}}$ has a final object $C$, then (7.9) admits a solution $\overline{f}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-limit diagram.

Proof. Using Proposition 7.2.2.9 and Corollary 4.6.6.25, we can replace $\operatorname{\mathcal{C}}$ by the simplicial set $\{ C\} \simeq \Delta ^0$, in which case the desired result follows from our assumption that $U$ is a cartesian fibration (see Example 7.1.5.9). $\square$