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.

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

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

satisfying some equations (see ^[1] or ^[2])