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.

This is a companion discussion topic for the original entry at https://www.epatters.org/post/retrotransformations
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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?

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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?