Is it true that in the category $R$-Mod of $R$-modules (and $R$-module homomorphisms), the diagram $$ M \longrightarrow M \otimes_{R} N \longleftarrow N,$$ where the arrows are the maps $m \to m \otimes 1$, $n \to n \otimes 1$, is a coproduct diagram?
I know that for commutative rings instead of $R$-modules this is true, but I don't why this is different. A detailed explanation for a noob in category theory would be nice. (not hw, I would just like to get this straight, thank you very much).