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Proposition 5.3.6.6. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets, and let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ denote the direct image of $\operatorname{\mathcal{E}}$ along $U$ (see Construction 4.5.9.1). Then the projection map $\pi : \operatorname{Res}_{ \operatorname{\mathcal{D}}/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibration of simplicial sets. Moreover, an edge $e$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ is $\pi $-cartesian if and only if it satisfies the following condition:

  • Form a commutative diagram

    \[ \xymatrix@R =50pt@C=50pt{ \Delta ^{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\ar [r]^-{ V_{e} } \ar [d] & \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\ar [r]^-{ U_{e} } \ar [d] & \Delta ^1 \ar [d]^{ \pi (e) } \\ \operatorname{\mathcal{E}}\ar [r]^-{V} & \operatorname{\mathcal{D}}\ar [r]^-{U} & \operatorname{\mathcal{C}}, } \]

    so that the edge $e$ can be identified with a morphism of simplicial sets $F_{e}: \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ such that $V_{e} \circ F_{e}$ is the identity. Then the morphism $F_{e}$ carries $U_{e}$-cocartesian morphisms of $\Delta ^{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ to $V_{e}$-cartesian morphisms of $\Delta ^{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.

Proof of Proposition 5.3.6.6. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of simplicial sets. We first claim that the projection map $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$ is an inner fibration of $\infty $-categories. To prove this, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex; in particular, we can assume that $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are also $\infty $-categories (Remark 4.1.1.9), the functor $V$ is an isofibration (Proposition 5.1.4.8), and $U$ is exponentiable (Proposition 5.3.6.1). Applying Corollary 4.5.9.18, we deduce that $\pi $ is an isofibration of simplicial sets, and therefore an inner fibration (Remark 4.5.5.7).

Let us say that an edge $e$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ is special if it satisfies condition $(\ast )$ of Proposition 5.3.6.6. Lemma 5.3.6.7 guarantees that every special edge of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ is $\pi $-cartesian. Moreover, if $Y$ is a vertex of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ and $\overline{e}: \overline{X} \rightarrow \pi (Y)$ is an edge of $\operatorname{\mathcal{C}}$, then Lemma 5.3.6.8 guarantees that there exists a special edge $e: X \rightarrow Y$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ satisfying $\pi (e) = \overline{e}$. It follows that $\pi $ is a cartesian fibration of simplicial sets.

To complete the proof of Proposition 5.3.6.6, we must show that every $\pi $-cartesian edge $e: X \rightarrow Y$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ is special. Without loss of generality we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$ and that $\pi (e)$ is the nondegenerate edge of $\operatorname{\mathcal{C}}$, so that we can identify $e$ with a functor $E: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ satisfying $V \circ E = \operatorname{id}_{\operatorname{\mathcal{D}}}$. Using Lemma 5.3.6.8, we can choose a special edge $e': X' \rightarrow Y$ of $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ satisfying $\pi (e') = \pi (e)$, corresponding to another functor $E': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. Since $e'$ is also $\pi $-cartesian, it is isomorphic to $e$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \Delta ^1 }( \Delta ^1, \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) )$, so that $E'$ is isomorphic to $E$ as an object of the $\infty $-category $\operatorname{Fun}_{ /\operatorname{\mathcal{D}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$. If $u$ is a $U$-cocartesian edge of $\operatorname{\mathcal{D}}$, then $E(u)$ is isomorphic to the $V$-cartesian morphism $E'(u)$ (as an object of the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{E}})$), and is therefore also $V$-cartesian (Corollary 5.1.2.5). $\square$