Kerodon

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

Proposition 4.2.6.1 (Joyal [MR1935979]). Let $K$ be a simplicial set, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets, and the projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration of simplicial sets. In particular, the simplicial sets $\operatorname{\mathcal{C}}_{f/}$ and $\operatorname{\mathcal{C}}_{/f}$ are $\infty $-categories (see Remark 4.1.0.4).

Proof of Proposition 4.2.6.1. Apply Corollary 4.2.6.5 in the special case $K_0 = \emptyset $. $\square$