Retrotransformations between lax double functors are introduced as the “multi-object” analogue of a cofunctor between categories. Notions of “monoidal cofunctor” between monoidal categories and of “multicofunctor” between multicategories are then derived as special cases.
It seems to me that a retrotransformation F\nRightarrow G:\mathbb{D}\to \mathbb{E}
might be equivalent to a transformation F^{ret}\Rightarrow G^{ret}:\mathbb{D}^{ret}\to \mathbb{E}^{ret}.
But I’m not sure yet. Do you think this could be true?
Good question! I hadn’t thought about this. Squinting at it a bit, it seems like it could be true. To get it straight, I would have to understand how the retrocell construction is functorial, so that I know how to get a double functor F^{\mathrm{ret}} from a double functor F. Is that in the literature or is it something that also needs to be worked out?