Kerodon

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

Remark 4.1.1.13. Let $q: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $q$ is an inner fibration.

$(2)$

For every simplex $\sigma : \Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ is an inner fibration.

$(3)$

For every simplex $\sigma : \Delta ^ n \rightarrow S$, the fiber product $\Delta ^ n \times _{S} X$ is an $\infty $-category.

The equivalence $(1) \Leftrightarrow (2)$ is immediate from the definition, and the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 4.1.1.10.