Kerodon

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

Corollary 7.1.7.20. 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. If $\overline{f}$ is a $U$-colimit diagram in $\operatorname{\mathcal{C}}$, then it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D}$. The converse holds if $U$ is a cartesian fibration.

Proof. Apply Proposition 7.1.7.19 in the special case $\operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}' = \{ D\} $. $\square$