Double Categories and (Op)Lax Morphisms

Nathanael Arkor told me this fun fact on [2024-11-19 Tue]:

My impression is that there is only one sensible notion of monoidal profunctor, when one has a double categorical perspective, and these turn out to be the lax monoidal functors into presheaf categories equipped with the convolution monoidal structure. Monoidal profunctors then subsume both lax monoidal functors and colax monoidal functors. In fact, if one considers the (virtual) double category of monoidal categories, strict monoidal functors, monoidal profunctors, and monoidal natural transformations, it is possible to extract (virtual) double categories of lax monoidal functors, colax monoidal functors, pseudo monoidal functors, etc.

I was meaning to write a little something about this, because I think it's not a perspective that has been properly appreciated in the literature: you can obtain the notion of "lax morphism" and "colax morphism" from double categorical structure "for free".

The key insight is that, given a functor f : A -> B, there is a bijection between lax monoidal structures on f and monoidal profunctor structures on the representable profunctor B(1, f) : A -|-> B; and dually, a bijection between colax monoidal structures on f and monoidal profunctor structures on the corepresentable profunctor B(f, 1) : B -|-> A.

A fun consequence of this theory is that you can obtain notions of lax morphism and colax morphism given any suitable functor of (virtual) double categories, which may not look like co/lax morphisms at first sight, but behave like them in an appropriate sense. For instance, there's a double functor for which the associated lax morphisms are functors between categories, and the associated colax morphisms are retrofunctors/cofunctors between categories.