2-functor

In mathematics, a 2-functor is a morphism between 2-categories. They may be defined formally using enrichment by saying that a 2-category is exactly a "Cat"-enriched category and a 2-functor is a "Cat"-functor.
Explicitly, if "C" and "D" are 2-categories then a 2-functor formula_1 consists of
such that each formula_5 strictly preserves identity objects and they commute with horizontal composition in "C" and "D".
See for more details and for lax versions.