Kerodon

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

Remark 5.4.7.6. The existence of morphisms between $(\infty ,2)$-categories which do not preserve thin $2$-simplices should be viewed as a feature of our formalism, rather than a bug. Recall that, if $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $2$-categories, then Theorem 2.3.4.1 supplies a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Strictly unitary lax functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \ar [d]^-{\sim } \\ \{ \textnormal{Morphisms of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$} \} . } \]

Consequently, we can think of general morphisms of simplicial sets as providing a generalization of the notion of (strictly) unitary lax functors to the setting of $(\infty ,2)$-categories.