# Kerodon

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Corollary 8.1.2.20. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, which we regard as an object of the twisted arrow $\infty$-category $\operatorname{Tw}(\operatorname{\mathcal{C}})$. Then:

• The morphism $f$ is $U$-cocartesian if and only if it is $V$-initial, where $V$ denotes the induced map

$\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \{ U(X) \} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}}).$
• The morphism $f$ is $U$-cartesian if and only if it is $V'$-initial, where $V'$ denotes the induced map

$\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \operatorname{Tw}(\operatorname{\mathcal{D}}) \times _{\operatorname{\mathcal{D}}} \{ U(Y) \} .$

Proof. We will prove the first assertion; the proof of the second is similar. Construction 8.1.2.7 supplies a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [d]^{ U_{X/} } \ar [r]^-{ \iota _{X} } & \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d]^{ V } \\ \operatorname{\mathcal{D}}_{U(X)/} \ar [r]^-{ \iota _{ U(X)} } & \{ U(X) \} \times _{ \operatorname{\mathcal{D}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{D}}), }$

where the horizontal maps are equivalences of $\infty$-categories (Proposition 8.1.2.9). By virtue of Remark 7.1.4.9, it will suffice to show that $f$ is $U$-cocartesian if and only if it is a $U_{X/}$-initial object of the $\infty$-category $\operatorname{\mathcal{C}}_{X/}$, which is a special case of Example 7.1.5.9. $\square$