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Example We define a strict $2$-category $\mathbf{Cat}$ as follows:

  • The objects of $\mathbf{Cat}$ are (small) categories.

  • For every pair of small categories $\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}\in \mathbf{Cat}$, we take $\underline{\operatorname{Hom}}_{ \mathbf{Cat} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ to be the category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.

  • The composition law on $\mathbf{Cat}$ is given by the usual composition of functors.

We will refer to $\mathbf{Cat}$ as the strict $2$-category of (small) categories. Note that the underlying ordinary category of $\mathbf{Cat}$ is the category $\operatorname{Cat}$ (whose objects are small categories and morphisms are functors).