2Category
A 2-category is a category enriched over Cat. It has objects, 1-morphisms between objects, and 2-morphisms between 1-morphisms. For any pair of objects A and B, the collection Hom(A, B) is itself a category. Its objects are 1-morphisms f: A → B, and its morphisms are 2-morphisms α: f ⇒ g between such 1-morphisms.
There is a composition operation that assigns to f: A → B and g: B → C a composite
Examples include the canonical 2-category Cat, whose objects are (small) categories, 1-morphisms are functors, and 2-morphisms
Relation to bicategories: Every 2-category is a bicategory with strict associativity and unit laws, but a bicategory