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Definition 2.2.5.5. We let $\operatorname{2Cat}_{\operatorname{Lax}}$ denote the ordinary category whose objects are (small) $2$-categories and whose morphisms are lax functors between $2$-categories (Definition 2.2.4.5), with composition given by Construction 2.2.5.1 and identity morphisms given by Example 2.2.5.4. We define (non-full) subcategories

\[ \operatorname{2Cat}_{\operatorname{Str}} \subsetneq \operatorname{2Cat}\subsetneq \operatorname{2Cat}_{\operatorname{Lax}} \supsetneq \operatorname{2Cat}_{\operatorname{ULax}} \]

  • The objects of $\operatorname{2Cat}$ are $2$-categories, and the morphisms of $\operatorname{2Cat}$ are functors.

  • The objects of $\operatorname{2Cat}_{\operatorname{Str}}$ are strict $2$-categories, and the morphisms of $\operatorname{2Cat}_{\operatorname{Str}}$ are strict functors.

  • The objects of $\operatorname{2Cat}_{\operatorname{ULax}}$ are $2$-categories, and the morphisms of $\operatorname{2Cat}_{\operatorname{ULax}}$ are strictly unitary lax functors.

We will refer to $\operatorname{2Cat}$ as the category of $2$-categories, and to $\operatorname{2Cat}_{\operatorname{Str}}$ as the category of strict $2$-categories.