Consider the usual tensor product of modules.
What is a good generalization of this tensor product?
The answer here is not supposed to be "a monoidal structure", I want the gist of this particular tensor product.
I know (vaguely) that there is such a tensor product for every category of $\mathsf{Set}$-models of a commutative algebraic theory. I wouldn't be surprised, if there was also one for commutative monoids in a symmetric monoidal category (with some properties).
But specifically, I want to take products also of things like ordered commutative monoids and similar variations of this theme (ideally, I have one construction that works for all these things).