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 $F:C\to D$ consists of

• a function $F:\text{Ob}C\to \text{Ob}D$, and
• for each pair of objects $c,c^{\prime }\in \text{Ob}C$, a functor $F_{c,c^{\prime }}:{\text{Hom}}_C(c,c^{\prime })\to {\text{Hom}}_D(Fc,F{c}^{\prime })$

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.