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.