Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 8.1.2.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the coslice inclusion $\iota _{X}$ restricts to a homotopy equivalence of Kan complexes

\[ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} . \]

Proof. Combine Proposition 8.1.2.5 with Corollary 5.1.5.4. $\square$