I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain complexes such that the identities between cup and cap products hold.
I was thinking about the comonoid objects in the closed monoidal categories, where you can create caps and cups, but to have standart $identities$ between caps and cups as is in the situation of the singular chain complexes it is not enough to have triangle and cocomutativity coherence conditions for the comonoid object $C$. So I am interested to know better framework.