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Corollary 5.1.5.10. Let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets. The following conditions are equivalent:

  • The morphism $q$ is a cartesian fibration.

  • Every locally $q$-cartesian edge of $X$ is $q$-cartesian.

  • For every $2$-simplex $\sigma $:

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

    of the simplicial set $X$, if $f$ and $g$ are locally $q$-cartesian, then $h$ is locally $q$-cartesian.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 5.1.4.5, the implication $(2) \Rightarrow (1)$ is immediate, and the equivalence $(2) \Leftrightarrow (3)$ follows from Proposition 5.1.5.9. $\square$