Kerodon

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Lemma 5.2.4.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $\pi : \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\rightarrow \Delta ^1$ be the projection map. Let $X$ be an object of $\operatorname{\mathcal{C}}$, let $Y$ be an object of $\operatorname{\mathcal{D}}$, and let $f: X \rightarrow Y$ be a morphism in the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$, which we identify with a morphism $e: F(X) \rightarrow Y$ in the $\infty$-category $\operatorname{\mathcal{D}}$ (see Remark 5.2.4.14). Then $f$ is $\pi$-cocartesian if and only if $e$ is an isomorphism in $\operatorname{\mathcal{D}}$.

Proof. Let $n \geq 2$ be an integer. Unwinding the definitions, we see that the datum of a lifting problem

5.19
$$\begin{gathered}\label{equation:cocartesian-in-relative-join} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{0} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\ar [d]^-{\pi } \\ \Delta ^{n} \ar@ {-->}[ur]^{\sigma } \ar [r] & \Delta ^1 } \end{gathered}$$

satisfying $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) } = f$ is equivalent to the datum of a diagram $\tau _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{D}}$ satisfying $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} )} = e$ (see Remark 5.2.4.2). Moreover, the lifting problem (5.19) admits a solution if and only if the corresponding map $\tau _0: \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{D}}$ can be extended to an $n$-simplex of $\operatorname{\mathcal{D}}$. The desired equivalence now follows from the characterization of isomorphisms given in Theorem 4.4.2.6 $\square$