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Lemma Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d]^{U} & \operatorname{\mathcal{D}}\ar [d]^{V} \\ \operatorname{\mathcal{C}}' \ar [r]^-{F'} & \operatorname{\mathcal{D}}', } \]

so that $U$ and $V$ induce a morphism $W: \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}'$. Let $e$ be an edge of the simplicial set $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying the following conditions:


If $e$ is contained in $\operatorname{\mathcal{C}}$, then it is $U$-cocartesian when viewed as an edge of $\operatorname{\mathcal{C}}$.


The image of $e$ under the map

\[ \rho : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\simeq \Delta ^1 \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}} \]

is $V$-cocartesian.

Then $e$ is $W$-cocartesian.

Proof. Let $n \geq 2$ be an integer and suppose we are given a lifting problem

\begin{equation} \begin{gathered}\label{equation:cocartesian-in-relative-join} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d]^-{W} \\ \Delta ^{n} \ar@ {-->}[ur]^{\sigma } \ar [r]^-{ \sigma ' } & \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}',} \end{gathered} \end{equation}

where $\sigma _0$ carries $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^{n}_{0}$ to the edge $e$. If $\sigma '$ is contained in the simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}'$, then the lifting problem (5.23) admits a solution by virtue of assumption $(1)$. Let us therefore assume that $\sigma '$ is not contained in $\operatorname{\mathcal{C}}'$. Let $\rho : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ be as in $(2)$, and define $\rho ': \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}' \rightarrow \operatorname{\mathcal{D}}'$ similarly. Unwinding the definitions, we can rewrite (5.23) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\rho \circ \sigma _0} \ar [d] & \operatorname{\mathcal{D}}\ar [d]^-{V} \\ \Delta ^{n} \ar@ {-->}[ur] \ar [r]^-{\rho ' \circ \sigma ' } & \operatorname{\mathcal{D}}',} \]

which admits a solution by virtue of assumption $(2)$. $\square$