I think that this sort of forward backward thing is a very general idea. There’s a one to many relationship called the adjoint, and a many to one relationship called the norm.
I wrote something about this here https://github.com/adamnemecek/adjoint
I wrote something about this here https://github.com/adamnemecek/adjoint
[0] https://arxiv.org/abs/2006.01631 [1] https://arxiv.org/abs/2101.10483 [2] https://arxiv.org/abs/1711.10455
I'm working on it; I'll send you an e-mail. Things quickly turned out to be more general than I realized last year.