Yoneda lemma and representation theorem for double categories

We study (vertically) normal lax double functors valued in the weak double category $\mathbb{C}\mathrm{at}$ of small categories, functors, profunctors and natural transformations, which we refer to as lax double presheaves. We show that for the theory of double categories they play a similar role as 2-functors valued in $\mathrm{Cat}$ for 2-categories. We first introduce representable lax double presheaves and establish a Yoneda lemma. Then we build a Grothendieck construction which gives a 2-equivalence between lax double presheaves and discrete double fibrations over a fixed double category. Finally, we prove a representation theorem showing that a lax double presheaf is represented by an object if and only if its Grothendieck construction has a double terminal object.