Kerodon

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

Example 5.1.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Applying Proposition 5.1.6.1 in the case where both $F$ and $G$ are the identity functor $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$, we deduce that the evaluation functor

\[ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]

is a cartesian fibration of $\infty $-categories, and the evaluation functor

\[ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}} \]

is a cocartesian fibration of $\infty $-categories.