Kerodon

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

Remark 5.4.7.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category. Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ takes values in the pith $\operatorname{Pith}(\operatorname{\mathcal{D}}) \subseteq \operatorname{\mathcal{D}}$. Consequently, the inclusion $\operatorname{Pith}(\operatorname{\mathcal{D}}) \hookrightarrow \operatorname{\mathcal{D}}$ induces a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors of $\infty $-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{Pith}(\operatorname{\mathcal{D}})$} \} \ar [d] \\ \{ \textnormal{Functors of $(\infty ,2)$-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$} \} . } \]

Note that this property (together with Proposition 5.4.5.6) characterize the simplicial set $\operatorname{Pith}(\operatorname{\mathcal{D}})$ up to unique isomorphism.