2-functor

In mathematics, specifically, in category theory, 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 $$F\colon C\to D$$ consists of such that each $$F_{c,c'}$$ strictly preserves identity objects and they commute with horizontal composition in C and D.
 * a function $$F\colon \text{Ob} C\to \text{Ob} D$$, and
 * for each pair of objects $$c,c'\in\text{Ob} C$$, a functor $$F_{c,c'}\colon \text{Hom}_{C}(c,c')\to\text{Hom}_D(Fc,Fc')$$

See for more details and for lax versions.