Kerodon

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

Remark 2.2.5.8. Let $\mathrm{MonCat}$ denote the ordinary category whose objects are monoidal categories and whose morphisms are monoidal functors (that is, the underlying category of the strict $2$-category $\mathbf{MonCat}$ of Example 2.2.0.5). Then the construction $\operatorname{\mathcal{C}}\mapsto B\operatorname{\mathcal{C}}$ determines a fully faithful embedding from $\mathrm{MonCat}$ to the category $\operatorname{2Cat}$ of Definition 2.2.5.5, which fits into a pullback diagram

\[ \xymatrix { \mathrm{MonCat} \ar [r]^-{ \operatorname{\mathcal{C}}\mapsto B\operatorname{\mathcal{C}}} \ar [d] & \operatorname{2Cat}\ar [d]^{ \operatorname{\mathcal{C}}\mapsto \operatorname{Ob}(\operatorname{\mathcal{C}}) } \\ \{ \ast \} \ar [r] & \operatorname{Set}; } \]

here $\ast = \{ X \} $ denotes a set containing a single fixed object $X$. Similarly, the ordinary category of monoidal categories and lax monoidal functors can be regarded as a full subcategory of $\operatorname{2Cat}_{\operatorname{Lax}}$.