Remark 5.3.1.12 (03LM)

Remark 5.3.1.12. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix { \operatorname{\mathcal{E}}' \ar [dr]_{U'} \ar [rr]^{ F } & & \operatorname{\mathcal{E}}\ar [dl]^{U} \\ & \operatorname{\mathcal{C}}, & } \]

where $U$ and $U'$ are cocartesian fibrations. Let $e: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$: For every $U'$-cocartesian edge $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ of $\operatorname{\mathcal{E}}'$ satisfying $U'( \widetilde{e} ) = e$, the image $F( \widetilde{e} )$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
$(2)$: For every vertex $\widetilde{X}$ of $\operatorname{\mathcal{E}}'$ satisfying $U'( \widetilde{X} ) = X$, there exists a $U'$-cocartesian edge $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ of $\operatorname{\mathcal{E}}'$ such that $F( \widetilde{e} )$ is $U$-cocartesian and $U'( \widetilde{e} ) = e$.

The implication $(1) \Rightarrow (2)$ is immediate from the definitions, and the implication $(2) \Rightarrow (1)$ follows from Remark 5.1.3.8.

Let $W$ be the collection of edges of $\operatorname{\mathcal{C}}$ which satisfy these conditions. Then $W$ contains all degenerate edges of $\operatorname{\mathcal{C}}$ and is closed under composition: that is, for every $2$-simplex

\[ \xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{ e' } & \\ X \ar [ur]^{e} \ar [rr]^{e''} & & Z } \]

of $\operatorname{\mathcal{C}}$, if $e$ and $e'$ belong to $W$, then $e''$ also belongs to $W$ (see Proposition 5.1.4.12).