Example 2.2.0.4. 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).