# Kerodon

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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.