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.
3. ^ 2-functor at the nLab