# Every under category with pushouts is monoidal

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.