Proposition 5.1.5.19. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ X \ar [dr]_{p} \ar [rr]^{r} & & Y \ar [dl]^{q} \\ & S & } \]
satisfying the following conditions:
The maps $p$ and $q$ are locally cartesian fibrations, and $r$ is an inner fibration.
The map $r$ carries locally $p$-cartesian edges of $X$ to locally $q$-cartesian edges of $Y$.
For every vertex $s$ of $S$, the induced map $r_{s}: X_{s} \rightarrow Y_{s}$ is a locally cartesian fibration.
Then $r$ is a locally cartesian fibration.
Proof of Proposition 5.1.5.19.
Choose a vertex $z \in X$ and an edge $\overline{h}: \overline{x} \rightarrow r(z)$ of the simplicial set $Y$. We wish to prove that there exists a locally $r$-cartesian edge $h: x \rightarrow z$ satisfying $r(h) = \overline{h}$. Since $p$ is a locally cartesian fibration, we can choose a locally $p$-cartesian edge $g: y \rightarrow z$ satisfying $p(g) = q(\overline{h})$. Assumption $(2)$ guarantees that $r(g)$ is locally $q$-cartesian, so we can choose a $2$-simplex $\overline{\sigma }$ of $Y$ satisfying
\[ d^{2}_0( \overline{\sigma } ) = r(g) \quad \quad d^{2}_1( \overline{\sigma } ) = \overline{h} \quad \quad q( \overline{\sigma } ) = s^{1}_0( q(\overline{h} ) ), \]
as indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & r(y) \ar [dr]^{ r(g) } & \\ \overline{x} \ar [ur]^{\overline{f}} \ar [rr]^{\overline{h} } & & r(z). } \]
Set $s = q( \overline{x} )$, so that $\overline{f}$ can be regarded as an edge of the simplicial set $Y_{s}$. Invoking assumption $(3)$, we conclude that there exists a locally $r_{s}$-cartesian edge $f: x \rightarrow y$ of $X_{s}$ satisfying $r(f) = \overline{f}$. Since $r$ is an inner fibration, we can choose a $2$-simplex $\sigma $ of $X$ satisfying
\[ d^{2}_0( \sigma ) = g \quad \quad d^{2}_2(\sigma ) = f \quad \quad r(\sigma ) = \overline{\sigma }, \]
as depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & y \ar [dr]^{ g } & \\ x \ar [ur]^{f} \ar [rr]^{h} & & z. } \]
We will complete the proof by showing that $h$ is locally $r$-cartesian. To prove this, we can replace $X$ and $Y$ by their pullbacks along the edge $\Delta ^1 \xrightarrow { q(\overline{h})} S$, and thereby reduce to the case $S = \Delta ^1$. In this case, the morphisms $p$ and $q$ are cartesian fibration (Remark 5.1.5.6), so that $g$ is $p$-cartesian and $r(g)$ is $q$-cartesian (Remark 5.1.4.5). Applying Corollary 5.1.2.6, we conclude that $g$ is $r$-cartesian. It follows from Remark 5.1.3.5 that $f$ is locally $r$-cartesian, so that $h$ is locally $r$-cartesian by virtue of Proposition 5.1.3.7.
$\square$