Proposition 8.6.1.13. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration, and suppose we are given a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{ \lambda _{+} } & \operatorname{\mathcal{C}}. } \]
Then $T$ satisfies condition $(2)$ of Definition 8.6.1.1 if and only if it satisfies both of the following conditions:
- $(2')$
For every object $Y \in \operatorname{\mathcal{E}}^{\dagger }$ having image $\overline{Y} = U^{\dagger }(Y)$ and every edge $e: \overline{Y} \rightarrow \overline{X}$ of $\operatorname{\mathcal{C}}$, the morphism $T$ carries $( \operatorname{id}_{Y}, e_{L} ): (Y, \operatorname{id}_{\overline{Y}} ) \rightarrow (Y, e)$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
- $(2'')$
Let $f: X \rightarrow Y$ be a $U^{\dagger }$-cartesian edge of $\operatorname{\mathcal{E}}^{\dagger }$. Set $\overline{X} = U^{\dagger }(X)$ and $\overline{Y} = U^{\dagger }(Y)$, so that $U^{\dagger }(f)$ can be identified with an edge $e: \overline{Y} \rightarrow \overline{X}$ of the simplicial set $\operatorname{\mathcal{C}}$. Then $T$ carries $( f, e_{R} ): (X, \operatorname{id}_{ \overline{X} }) \rightarrow (Y, e )$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{ \overline{Y} }$.
Proof.
The implication $(2) \Rightarrow (2')$ is immediate from the definitions, and the implication $(2) \Rightarrow (2'')$ follows from Proposition 5.1.4.12. For the converse, suppose that conditions $(2')$ and $(2'')$ are satisfied. Let $f: X \rightarrow Y$ be a $U^{\dagger }$-cartesian edge of $\operatorname{\mathcal{E}}^{\dagger }$, and let us identify $U^{\dagger }(f)$ with an edge $e: \overline{Y} \rightarrow \overline{X}$ of the simplicial set $\operatorname{\mathcal{C}}$. Suppose we are given a lift of $e$ to an edge $\widetilde{e}: u \rightarrow v$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$, which we identify with a $3$-simplex $\sigma : \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$ depicted in the diagram
\[ \xymatrix@C =50pt@R=50pt{ \overline{X} \ar [d]^{u} & \overline{Y} \ar [l]_{e} \ar [d]^{v} \\ \overline{X}' \ar [r] & \overline{Y}'. } \]
We wish to show that $T( f, \widetilde{e} )$ is a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
Let $\sigma '$ denote the degenerate $5$-simplex of $\operatorname{\mathcal{C}}$ given by $\gamma ^{\ast }(\sigma )$, where $\gamma : [5] \rightarrow [3]$ is given by $\gamma (0) = 0$, $\gamma (1) = \gamma (2) = \gamma (3) = 1$, $\gamma (4) = 2$, and $\gamma (5) = 3$. Let us abuse notation by identifying $\sigma '$ with the $2$-simplex of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ depicted in the diagram
8.75
\begin{equation} \begin{gathered}\label{equation:check-second-condition} \xymatrix@C =50pt@R=50pt{ \overline{X} \ar [d]^{\operatorname{id}} & \overline{X} \ar [l]_{\operatorname{id}} \ar [d]^{u} & \overline{Y} \ar [l]_{e} \ar [d]^{v} \\ \overline{X} \ar [r]^-{u} & \overline{X}' \ar [r] & \overline{Y}'. } \end{gathered} \end{equation}
Evaluating $T$ on the pair $(s^{1}_0(f), \sigma ')$, we obtain a $2$-simplex of $\operatorname{\mathcal{E}}$ depicted in the diagram
\[ \xymatrix@C =50pt@R=50pt{ & T(X, u ) \ar [dr]^{ T(f, \widetilde{e} ) } & \\ T(X, \operatorname{id}_ X) \ar [ur]^{ T( \operatorname{id}_{X}, u_{L})} \ar [rr] & & T( Y, v), } \]
where the left diagonal map is $U$-cocartesian by virtue of assumption $(2')$. Consequently, to show that the $T( f, \widetilde{e} )$ is $U$-cocartesian, it will suffice to show that the horizontal edge is $U$-cocartesian (Proposition 5.1.4.13). Note that, in the special case where $\sigma = s^{2}_0( \sigma _0 )$ for some $2$-simplex $\sigma _0$ of $\operatorname{\mathcal{C}}$, the horizontal edge coincides with $T( \operatorname{id}_{X}, v_ L )$, which is also $U$-cocartesian by virtue of $(2')$.
To handle the general case, we can replace $\sigma $ by the $3$-simplex of $\operatorname{\mathcal{C}}$ given by the outer rectangle of the diagram (8.75) (that is, by the $3$-simplex $\sigma '|_{ \operatorname{N}_{\bullet }( \{ 0 < 2 < 3 < 5 \} ) }$, and thereby reduce to the special case where $\sigma = s^{2}_1( \sigma _1)$, for some $2$-simplex $\sigma _1$ of $\operatorname{\mathcal{C}}$ (so that $\overline{X} = \overline{X}'$ and $u$ is a degenerate edge of $\operatorname{\mathcal{C}}$). In this case, we let $\sigma ''$ denote the $5$-simplex of $\operatorname{\mathcal{C}}$ given by $\beta ^{\ast }(\sigma _1)$, where $\beta : [5] \rightarrow [2]$ is given by $\beta (0) =\beta (1) = 0$, $\beta (2) = \beta (3) = \beta (4) = 1$, and $\beta (5) = 2$. Let us view $\sigma ''$ as a $2$-simplex of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ depicted in the diagram
\[ \xymatrix@C =50pt@R=50pt{ \overline{X} \ar [d]^{u} & \overline{Y} \ar [l]_{e} & \overline{Y} \ar [l]_{\operatorname{id}} \\ \overline{X}' \ar [r]^-{\operatorname{id}} & \overline{X}' \ar [r] & \overline{Y}'. } \]
Evaluating $T$ on the pair $(s^{1}_1(f), \sigma '')$, we obtain a $2$-simplex of $\operatorname{\mathcal{E}}$ depicted in the diagram
\[ \xymatrix@C =50pt@R=50pt{ & T(Y, e) \ar [dr] & \\ T(X, u) \ar [ur]^{ T(f,e_{R}) } \ar [rr]^{ T( f, \widetilde{e} ) } & & T( Y, v). } \]
Here the left diagonal edge is $U$-cocartesian by virtue of assumption $(2'')$ and the right diagonal edge is $U$-cocartesian by virtue of the special case treated above. Applying Proposition 5.1.4.13, we conclude that $T( f, \widetilde{e} )$ is also $U$-cocartesian.
$\square$