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An ∞-cosmos is a setting in which to develop the formal category theory of (∞, 1)-categories. In this paper, we explore a few atypical examples of ∞-cosmoi whose objects are 2-categories or bicategories rather than (∞, 1)categories and compare the formal category theory that is so-encoded with classical 2-category and bicategory theory. We hope this work will inform future explorations into formal (∞, 2)-category theory. Contents 1. Introduction 1 2. ∞-cosmoi of 2-categories and bicategories 2 3. Formal category theory in an ∞-cosmos 6 4. The formal theory of 2-categories, 2-functors, and 2-natural transformations 8 5. The formal theory of bicategories, normal pseudofunctors, and icons 12 6. Another ∞-cosmos of bicategories 18 References 21