# 2-functor

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 $\displaystyle{ F\colon C\to D }$ consists of

• a function $\displaystyle{ F\colon \text{Ob} C\to \text{Ob} D }$, and
• for each pair of objects $\displaystyle{ c,c'\in\text{Ob} C }$, a functor $\displaystyle{ F_{c,c'}\colon \text{Hom}_{C}(c,c')\to\text{Hom}_D(Fc,Fc') }$

such that each $\displaystyle{ F_{c,c} }$ strictly preserves identity objects and they commute with horizontal composition in C and D.

See [3] for more details and for lax versions.

## 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.