2-functor explained

In mathematics, specifically, in category theory, a 2-functor is a morphism between 2-categories.^[1] 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.^[2]

Explicitly, if C and D are 2-categories then a 2-functor

F\colon C\to D consists of

• a function

F\colon Ob C\to Ob D , and

• for each pair of objects

c,c'\in Ob C , a functor F _{c,c'} \colon Hom _{C} (c,c')\to Hom _{D(Fc,Fc')} such that each F _{c,c'} strictly preserves identity objects and they commute with horizontal composition in C and D.

See for more details and for lax versions.

Notes and References

1. Kelly . G.M. . Street . R. . 1974 . Review of the elements of 2-categories . Category Seminar . 420 . 75–103 .
2. G. M. Kelly. Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, (10), 2005.