Lemma 5.6.8.5. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{ \widetilde{\mathscr {F} }} \ar [d]^{U} & \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F} } & \operatorname{\mathcal{QC}}, } \]
where $U$ is a cocartesian fibration. Let $j: \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ be an inner anodyne morphism of simplicial sets, let $\operatorname{\mathcal{E}}_0$ denote the fiber product $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$, and let $U_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{C}}_0$ denote the projection map. If $\widetilde{\mathscr {F}}|_{ \operatorname{\mathcal{E}}_0}$ witnesses $\mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ as a covariant transport representation for $U_0$, then $\widehat{\mathscr {F}}$ witnesses $\mathscr {F}$ as a covariant transport representation for $U$.
Proof.
Let $S$ denote the collection of all morphisms of simplicial sets $i: A \rightarrow B$ with the following property: for every morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{C}}$, if the restriction $\widetilde{\mathscr {F}}|_{ A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}}$ witnesses $\mathscr {F}|_{ A }$ as a covariant transport representation for the projection map $A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow A$, then $\widetilde{\mathscr {F}}|_{ B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}}$ witnesses $\mathscr {F}|_{ B }$ as a covariant transport representation for the projection map $B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow B$. To prove Lemma 5.6.8.5, it will suffice to show that every inner anodyne morphism of simplicial sets belongs to $S$. It is not difficult to see that the collection of morphisms $S$ is weakly saturated, in the sense of Definition 1.5.4.12. It will therefore suffice to show that, for every pair of integers $0 < i < n$, the inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ belongs to $S$. We may therefore assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ and $\operatorname{\mathcal{C}}_0 = \Lambda ^{n}_{i}$ is an inner horn.
Since every vertex of $\Delta ^ n$ is contained in $\Lambda ^{n}_{i}$, it follows immediately that the pair $(\mathscr {F}, \widetilde{\mathscr {F}})$ satisfies condition $(a)$ of Remark 5.6.5.3. To verify $(b)$, let $e: X \rightarrow Z$ be an $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ having image $\overline{e} = U(e)$ in $\Delta ^ n$; we wish to show that $\widetilde{\mathscr {F}}(e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}'$. If $\overline{e}$ belongs to the horn $\Lambda ^ n_{i}$, then this follows from our assumption on $\widetilde{F}|_{ \operatorname{\mathcal{E}}_0 }$. We may therefore assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^{2}$ and that $\overline{e}: 0 \rightarrow 2$ is the “long” edge of the simplex $\Delta ^{2}$. Since $U$ is a cocartesian fibration, there exists a $U$-cocartesian edge $e': X \rightarrow Y$ of $\operatorname{\mathcal{E}}$, where $U(Y) = 1$. Our assumption that $e'$ is $U$-cocartesian guarantees the existence of a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{e''} & \\ X \ar [ur]^{ e'} \ar [rr]^-{e} & & Z } \]
of $\operatorname{\mathcal{E}}$, and Proposition 5.1.4.12 implies that $e''$ is also $U$-cocartesian. Since $\widetilde{\mathscr {F}}|_{ \operatorname{\mathcal{C}}_0}$ carries $U_0$-cocartesian morphisms of $\operatorname{\mathcal{E}}_0$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$, it follows that $\widetilde{\mathscr {F}}(e')$ and $\widetilde{\mathscr {F}}(e'')$ are $V$-cocartesian edges of $\operatorname{\mathcal{E}}'$. Applying Proposition 5.1.4.12 again, we deduce that $\widetilde{\mathscr {F}}(e)$ is also $V$-cocartesian.
$\square$