I am working with a category $\mathcal{C}$ in which the hom-sets are orders. Alternatively, we could look at it as a bicategory in which hom-sets for 2-cells are thin.
Is there a notion of monoidal category that has into account this order structure in a lax way? I'd like the tensor product $\otimes : \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ to be a lax functor instead of a strict/pseudo functor. I'd like to find what are the coherences one would need for that kind of structure.
It's not completely clear to me, but I don't think this would be equivalent to ask for $\mathcal{C}$ to be an order enriched monoidal category, would it?