Category of categories

A category of categories is a category $𝖢$ such that any two objects $𝖠, 𝖡 \in O b 𝖢$ are categories and the morphisms $𝖠 \to 𝖡$ are exactly the functors $𝖠 \to 𝖡$ . #m/def/cat/evil

Note that in general, equality of functors is ill-defined unless we have equality of objects. Therefore categories of categories are usually restricted to strict categories, see e.g. $𝖢 𝖺 𝗍$ . More general categories (without equality of objects) assemble into bicategories, see e.g. $ℭ 𝔞 𝔱$