Lax natural transformation

**Definition.** A *lax-natural transformation* $\alpha \colon F \Rightarrow G$ of 2-functors $F, G \colon K \to L$ consists of a family of arrows $\alpha_A \colon F A \to G A$ in $L$ indexed over the objects $A \in K$ together with, for each arrow $f$ of $K$, a distinguished 2-cell

$\alpha_f \colon Gf \circ \alpha_A \Rightarrow \alpha_B \circ Ff$

satisfying the two following coherence conditions.

- for any composable arrows $f$ and $g$ in $K$, $\alpha_f \ast \alpha_g = \alpha_{gf}$ where $\ast$ means pasting the two squares.
- for any 2-cell $\theta \colon f \Rightarrow g$ in $K$, $F\theta$ and $G\theta$ form the top and bottom of a commutative cylinder connecting the squares for $\alpha_f$ and $\alpha_g$

If the $\alpha_f$ are all invertible, $\alpha$ is called a *pseudonatural transformation*. If the $\alpha_f$ are all identities, $\alpha$ is called a *2-natural transformation*.

- J. Gray?, Formal Category Theory: Adjointness for 2-Categories, volume 391 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1974.

category: mathematical methods