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 consists of * a function , and * for each pair of objects a functor such that each strictly preserves identity objects and they commute with horizontal composition in C and D. See for more details and for lax versions.