2-functor

In the mathematical field of Category theory, a 2-functor is a morphism between 2-categories. Because strict 2-categories can be defined as categories enriched in Cat, the category of small categories, a 2-functor can be defined succinctly as a Cat-enriched functor.

Spelling this out a bit, let C and D be categories. A 2-functor $F\colon C\to D$ consists of

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

such that these functors strictly preserves identity objects and commute with compositions.

See ^[1] for more details and for lax versions.

References

1. 2-functor in nLab