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 [math]\displaystyle{ F,G\colon C\to D }[/math] be 2-functors. A lax natural transformation [math]\displaystyle{ \alpha\colon F\to G }[/math] between them consists of

• a morphism [math]\displaystyle{ \alpha_c\colon F(c)\to G(c) }[/math] in D for every object [math]\displaystyle{ c\in C }[/math] and
• a 2-morphism [math]\displaystyle{ \alpha_f\colon G(f)\circ\alpha_c \to \alpha_{c'}\circ F(f) }[/math] for every morphism [math]\displaystyle{ f\colon c\to c' }[/math] in C

satisfying some equations (see or )