?? so ... comm monoid as one-object sym mon cat ... ?? functorial operation associated to span s under sym mon structure as suspiciously similar to ordinary operation associated to s under comm monoid structure ... ?? ...

?? bi-module manifestation f# of functor f : x -> y ... f#(x1,y1) = [f(x1),y1]_y ... ??? ....

?? s from a to b ... comm monoid m .... f : m^a -> m^b associated to s under comm monoid / sym mon structure of m .... f#(*,*)=[f(*),*]_m^b .... ???? so m^b as m^a,m^b-bimodule .... vanilla (= "cayley" ?? ...) m^b-module self-structure, m^a-module structure given by matrix / span s ... ???

??? so ... ???? "cartesian" structure here ????? ..... ????? ......

?? adjoint bimodules here ???? ...... affine vs non-affine case ..... ???? ....

??? complex / symplectic // commutative / symmetric .... ???? ....