Example 2.2.0.5. We define a strict $2$-category $\mathbf{MonCat}$ as follows:
The objects of $\mathbf{MonCat}$ are (small) monoidal categories.
For every pair of small monoidal categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, we take $\underline{\operatorname{Hom}}_{ \mathbf{MonCat} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ to be the category $\operatorname{Fun}^{\otimes }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (Notation 2.1.6.9).
The composition law on $\mathbf{MonCat}$ is given by the composition of monoidal functors described in Remark 2.1.6.13.
There are several obvious variants on this construction: for example, we can work with nonunital monoidal categories in place of monoidal categories, or lax monoidal functors in place of monoidal functors.