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Example 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

  • The composition law on $\mathbf{MonCat}$ is given by the composition of monoidal functors described in Remark

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.