Let $𝐹,𝐺:\mathrm{ℬ︀}\to \mathrm{𝒞︀}$ be lax functors. A lax natural transformation $𝜎:𝐹\Rightarrow 𝐺$ consists of

1. for each 0-cell $𝑥$, a 1-cell ${𝜎}_{𝑥}:𝐹𝑥\to 𝐺𝑥$;

2. for each 1-cell $𝑓:𝑥\to 𝑦$, a 2-cell filling the naturality square,

   $${𝜎}_{𝑓}:𝐹𝑓\star {𝜎}_{𝑦}\Rightarrow {𝜎}_{𝑥}\star 𝐺𝑓;$$
3. such that ${𝜎}_{𝑓}$ is natural in $𝑓$: for a 2-cell $𝜃:𝑓\Rightarrow 𝑔$, $(𝐹𝜃▷{𝜎}_{𝑦})\star {𝜎}_{𝑔}={𝜎}_{𝑓}\star ({𝜎}_{𝑥}◁𝐺𝜃)$;
4. and such that $𝜎$ respects the comparison cells of $𝐹$ and $𝐺$: one law relating ${𝜎}_{1_{𝑥}}$ to ${𝐹}^0$, ${𝐺}^0$ and the unitors, and one relating ${𝜎}_{𝑓\star 𝑔}$ to ${𝜎}_{𝑓}$, ${𝜎}_{𝑔}$, ${𝐹}^2$, ${𝐺}^2$ and the associators.

A lax natural transformation is pseudonatural when every ${𝜎}_{𝑓}$ is invertible. As with pseudofunctors, this is a property, so the pseudonatural transformations $𝐹\Rightarrow 𝐺$ are a full subcategory of the category of lax transformations and modifications.

Between prestacks the 1-cells are taken to be pseudonatural. The reason is biuniversality: a transformation whose components ${𝜎}_{𝑥}$ are all equivalences of categories is an equivalence of prestacks only if its naturality cells are invertible, and a biuniversal element should be exactly a representation of a prestack up to such an equivalence.