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Lemma 5.2.4.13. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{C}}\ar [r] \ar [d]^{U} & \operatorname{\mathcal{E}}\ar [d]^{W} & \operatorname{\mathcal{D}}\ar [d]^{V} \ar [l] \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{E}}' & \operatorname{\mathcal{D}}' \ar [l] } \]

in which the vertical morphisms are inner fibrations. Then the induced map

\[ F: \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{D}}' \]

is also an inner fibration.

Proof. Unwinding the definitions, we see that $F$ factors as a composition

\[ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{D}}\xrightarrow {H} \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{D}}', \]

where $G$ is a pullback of the inner fibration $\operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}\star _{\operatorname{\mathcal{E}}'} \operatorname{\mathcal{E}}$ of Lemma 5.2.4.12 and $H$ is a pullback of the inner fibration $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star \operatorname{\mathcal{D}}'$ of Proposition 4.3.3.23. $\square$