Maybe a stupid question, but I can't figure it out.
Suppose we have some category $C$ where all pushouts exist, $c\in Ob C$ and $c\downarrow C$ an under category (i.e. a category with objects of the form $c\to d$ and obvious morphisms). One can define a "tensor product" by the following formula: $a\otimes b := a* _c b$ and the identity element by $c\xrightarrow{id} c$.
Is it true, that the category $c\downarrow C$ with this "tensor product" monoidal? It seems that all the conditions are satisfied: this category has the identity element, pushout is associative and so on.
Coproducts and initial objects in any category form a monoidal structure on that category. Coproducts in under-categories are pushouts in the underlying category. The identity arrow is the initial object of the under-category. Thus the under-category has all finite coproducts if the underlying category has pushouts.