2-functor

In mathematics, 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 $F\colon \text C\to \text D$, and
* for each pair of objects $c,c'\in C$ a functor $F_\colon \text_(c,c')\to\text_D(Fc,Fc')$
such that each $F_$ strictly preserves identity objects and they commute with horizontal composition in ''C'' and ''D''.

See for more details and for lax versions.

References

Functors
Higher category theory

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