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

• 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 $F_{c,c'}\colon {\text{Hom}}_{C}(c,c')\to {\text{Hom}}_{D}(Fc,Fc')$

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.