# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example 5.2.3.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Applying Lemma 5.2.3.17 to the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{F} \ar [d] & \operatorname{\mathcal{D}}\ar [d] \\ \Delta ^{0} \ar@ {=}[r] & \Delta ^{0}, }$

we deduce that the projection map

$\pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^{0} \star _{\Delta ^{0} } \Delta ^{0} \simeq \Delta ^1$

is a cocartesian fibration. Moreover, a morphism $e: X \rightarrow Y$ of the $\infty$-category $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ is $\pi$-cocartesian if and only if it satisfies one of the following three conditions:

• The objects $X$ and $Y$ belong to $\operatorname{\mathcal{C}}$ and $e$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$.

• The objects $X$ and $Y$ belong to $\operatorname{\mathcal{D}}$ and $e$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$.

• The object $X$ belongs to $\operatorname{\mathcal{C}}$, the object $Y$ belongs to $\operatorname{\mathcal{D}}$, and $e$ corresponds to an isomorphism $e_0: F(X) \rightarrow Y$ in the $\infty$-category $\operatorname{\mathcal{D}}$ (under the identification of Remark 5.2.3.14).