Lax natural transformation
# Lax natural transformation

In the mathematical field of category theory, specifically the theory of 2-categories, a lax natural transformation is a kind of morphism between 2-functors.

## Definition

Let C and D be 2-categories, and let $F,G\colon C\to D$ be 2-functors. A lax natural transformation $\alpha \colon F\to G$ between them consists of

* a morphism $\alpha _{c}\colon F(c)\to G(c)$ in D for every object $c\in C$ and
* a 2-morphism $\alpha _{f}\colon G(f)\circ \alpha _{c}\to \alpha _{c'}\circ F(f)$ for every morphism $f\colon c\to c'$ in C

satisfying some equations (see or)

