Kerodon

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Proposition 5.1.3.7. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets and let $\sigma : \Delta ^2 \rightarrow X$ be a $2$-simplex of $X$, which we will depict as a diagram

\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]_{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z. } \]

  • Suppose that $g$ is $q$-cartesian. Then $f$ is locally $q$-cartesian if and only if $h$ is locally $q$-cartesian.

  • Suppose that $f$ is $q$-cocartesian. Then $g$ is locally $q$-cocartesian if and only if $h$ is locally $q$-cocartesian.

Proof. We will prove the first assertion; the proof of the second is similar. Using Remarks 5.1.1.11 and 5.1.3.5, we can replace $q$ by the projection map $\Delta ^2 \times _{S} X \rightarrow \Delta ^2$, and thereby reduce to the case where $S = \Delta ^2$ and $q(\sigma )$ is the identity morphism $\operatorname{id}_{ \Delta ^2 }$. In this case, both $X$ and $S$ are $\infty $-categories and the morphisms $f$ and $h$ are locally $q$-cartesian if and only if they are $q$-cartesian (Remark 5.1.3.4). The desired result now follows from Corollary 5.1.2.4. $\square$