Definition

Given two 2-functors $U, V: S \stackrel{\to}{\to} C$ between 2-categories, a pseudonatural transformation $\phi: U \to V$ is a rule that assigns to each object $s$ of $S$ a morphism $\phi(s): U(s) \to V(s)$ of $C$, and to each morphism $f: r \to s$ of $S$ an invertible 2-morphism $\phi(f)$ of $C$:

such that the following coherence laws are satisfied in $C$ (throughout we leave the associators and unitors in $C$ implicit):

1. respect for composition: for all composable morphisms $r \stackrel{f}{\to} s \stackrel{g}{\to} t$ in $S$ we have an equality

   $$\array{ && U(s) \\ & {}^{\mathllap{U(f)}}\nearrow &\downarrow^{\phi(s)}& \searrow^{\mathrlap{U(g)}} \\ U(r) &\swArrow_{\phi(f)}&V(s) &\swArrow_{\phi(g)}& U(t) \\ {}^{\mathllap{\phi(r)}}\downarrow &{}^{V(f)}\nearrow& \Downarrow^{V(f,g)} &\searrow^{V(g)}& \downarrow^{\mathrlap{\phi(t)}} \\ V(r) &&\underset{V( g\circ f)}{\to}&& V(t) } \;\;\; = \;\;\; \array{ && U(s) \\ & {}^{\mathllap{U(f)}}\nearrow &\Downarrow^{U(f,g)}& \searrow^{\mathrlap{U(g)}} \\ U(r) &&\stackrel{U(g \circ f)}{\to}&& U(t) \\ {}^{\mathllap{\phi(r)}}\downarrow && \swArrow_{\phi(g \circ f )} && \downarrow^{\mathrlap{\phi(t)}} \\ V(r) &&\underset{V(g \circ f)}{\to}&& V(t) } \,,$$

   of pasting 2-morphisms as indicated, where $U(f,g)$ and $V(f,g)$ denote the compositors of the 2-functors $U$ and $V$,

2. respect for units, (…)

3. naturality

   for every 2-morphism

   $$\array{ && \stackrel{f}{\to} \\ & \nearrow && \searrow \\ r &&\Downarrow^{F}&& s \\ & \searrow && \nearrow \\ && \underset{g}{\to} }$$

   in $S$ an equality

   $$\array{ && \stackrel{U(f)}{\to} \\ & \nearrow &\Downarrow^{U(F)}& \searrow \\ U(r) &&\stackrel{U(g)}{\to}&& U(s) \\ {}^{\mathllap{\phi(r)}}\downarrow &&\swArrow_{\phi(g)}&& \downarrow^{\mathrlap{\phi(s)}} \\ V(r) &&\underset{V(g)}{\to}&& V(s) } \;\;\; = \;\;\; \array{ U(r) &&\stackrel{U(f)}{\to}&& U(s) \\ {}^{\mathllap{\phi(r)}}\downarrow &&\swArrow_{\phi(f)}&& \downarrow^{\mathrlap{\phi(s)}} \\ V(r) &&\stackrel{V(f)}{\to}&& V(s) \\ & \searrow &\Downarrow^{V(F)}& \nearrow \\ && \underset{V(g)}{\to} }$$

   in $C$.