# Kerodon

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Lemma 5.2.3.17. Suppose we are given a commutative diagram of simplicial sets

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

Suppose that $U$ and $V$ are cocartesian fibrations, and that the morphism $F$ carries $U$-cocartesian edges of $\operatorname{\mathcal{C}}$ to $V$-cocartesian edges of $\operatorname{\mathcal{D}}$. Then the induced map $W: \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}'$ is also a cocartesian fibration. Moreover, an edge $e$ of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $W$-cocartesian if and only if it satisfies conditions $(1)$ and $(2)$ of Lemma 5.2.3.16.

Proof. It follows from Lemma 5.2.3.12 that $W$ is an inner fibration of simplicial sets. Let us say that an edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special if it satisfies conditions $(1)$ and $(2)$ of Lemma 5.2.3.16, so that every special edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $W$-cocartesian. We consider three cases:

• Suppose that $X$ belongs to $\operatorname{\mathcal{C}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{C}}'$. In this case, our assumption that $U$ is a cocartesian fibration guarantees that we can lift $\overline{e}$ to $U$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$. Since $F(e)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{D}}$, the edge $e$ is special.

• Suppose that $X$ belongs to $\operatorname{\mathcal{C}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{D}}'$. In this case, we can identify $\overline{e}$ with an edge $\overline{e}_0: V( F(X) ) \rightarrow \overline{Y}$ of the simplicial set $\operatorname{\mathcal{D}}'$. Since $V$ is a cocartesian fibration, we can lift $\overline{e}_0$ to a $V$-cocartesian morphism $e_0: F(X) \rightarrow Y$ of $\operatorname{\mathcal{D}}$, which we can identify with a special edge $e: X \rightarrow Y$ of the simplicial set $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ satisfying $W(e) = \overline{e}$.

• Suppose that $X$ belongs to $\operatorname{\mathcal{D}}$ and $\overline{Y}$ belongs to $\operatorname{\mathcal{D}}'$. In this case, our assumption that $V$ is a cocartesian fibration guarantees that we can lift $\overline{e}$ to $V$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{D}}\subseteq \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, which is then special when regarded as an edge of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$.

To complete the proof, it will suffice to show that every $W$-cocartesian edge $e: X \rightarrow Y$ of $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is special. Applying the preceding argument, we can choose a special edge $e': X \rightarrow Y'$ satisfying $W(e') = W(e)$. Set $\overline{Y} = W(Y) = W(Y')$. Since $e$ and $e'$ are both $W$-cocartesian, Remark 5.1.3.8 supplies a $2$-simplex $\sigma$ of the simplicial set $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ with boundary given by

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

where $u$ is an isomorphism in the $\infty$-category $\{ \overline{Y} \} \times _{ ( \operatorname{\mathcal{C}}' \star _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{D}}' ) } (\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}})$. Applying Remark 5.1.3.8 to the cocartesian fibrations $U$ and $V$, we deduce that the edge $e$ is also special. $\square$