A lax functor $𝐹:\mathrm{ℬ︀}\to \mathrm{𝒞︀}$ between bicategories consists of

1. a map on 0-cells, $𝑥\mapsto 𝐹𝑥$;
2. for all $𝑥,𝑦$, a functor ${𝐹}_{𝑥,𝑦}:\mathrm{ℬ︀}(𝑥,𝑦)\to \mathrm{𝒞︀}(𝐹𝑥,𝐹𝑦)$, acting on 1-cells and 2-cells;
3. a unit comparison, natural 2-cells ${𝐹}_{𝑥}^0:1_{𝐹𝑥}\Rightarrow 𝐹(1_{𝑥})$;
4. a composition comparison, 2-cells ${𝐹}_{𝑓,𝑔}^2:𝐹𝑓\star 𝐹𝑔\Rightarrow 𝐹(𝑓\star 𝑔)$ natural in $𝑓$ and $𝑔$;

5. such that three coherence laws hold, one for each structure cell of $\mathrm{ℬ︀}$:

   • left unit: $({𝐹}^0▷𝐹𝑓)\star {𝐹}_{1,𝑓}^2\star 𝐹({𝜆}_{𝑓})={𝜆}_{𝐹𝑓}$;
   • right unit: $(𝐹𝑓◁{𝐹}^0)\star {𝐹}_{𝑓,1}^2\star 𝐹({𝜌}_{𝑓})={𝜌}_{𝐹𝑓}$;
   • associativity: $({𝐹}_{𝑓,𝑔}^2▷𝐹ℎ)\star {𝐹}_{𝑓\star 𝑔,ℎ}^2\star 𝐹({𝛼}_{𝑓,𝑔,ℎ})={𝛼}_{𝐹𝑓,𝐹𝑔,𝐹ℎ}\star (𝐹𝑓◁{𝐹}_{𝑔,ℎ}^2)\star {𝐹}_{𝑓,𝑔\star ℎ}^2$.

Here $\star$ between 2-cells is vertical composition, and $𝜃▷ℎ$ and $ℎ◁𝜃$ are whiskerings.

Lax functors compose: ${(𝐺\circ 𝐹)}^0={𝐺}^0\star 𝐺({𝐹}^0)$ and ${(𝐺\circ 𝐹)}_{𝑓,𝑔}^2={𝐺}_{𝐹𝑓,𝐹𝑔}^2\star 𝐺({𝐹}_{𝑓,𝑔}^2)$. The coherence laws of the composite follow from those of $𝐹$ and $𝐺$ and naturality of ${𝐺}^2$, without using the triangle or pentagon of any of the bicategories involved.

A lax functor whose comparison cells are invertible is a pseudofunctor.