Definition 1.2.5 (The 2-category of $U$-small categories).label Let $U$ be a Grothendieck universe then define $\text{CAT}_{U}$ to be the $2$-category whose:

• Objects are $U$-small categories, i.e. categories $\frak{C}$ such that $\ob{\frak{C}}\in U$ and $\mor{\frak{C}}{X }{Y }$ for all $X,Y\in\ob{\frak{C}}$.

• $1$-morphisms are functors between $U$-small categories.

• $2$-morphisms are natural transformations between such functors.