For a ring $R$, an $R$-right-module $M$, an $R$-left-module $N$, and an abelian group $P$, one can use the universal property of the tensor product to construct maps $$ M\otimes_R N\to P. $$ It concrete cases, it is often easy to see that the constructed map is surjective by just writing down pre-images.
It seems to be harder to verify that the map is injective, because then one has to consider general sums of elementary tensors in $M\otimes_R N$.
Are there any tricks to avoid this, and to achieve injectivity more elegantly?
More concretely, an example I have in mind is the following: I have a pre-additive category $C$ with finitely many objects. I consider modules over this category (that is, functors from $C$ to Abelian groups; or, equivalently, modules over the category ring of $C$). Now, I consider the "free $C$-right-module $Q_Y$ over an object $Y$ in $C$" which is the hom functor $C(\bullet,Y)$. I want to show that for an arbitrary left-module $M$: $$ Q_Y\otimes_C M\cong M(Y) $$ as abelian groups. Using the module structure of $M$, I can accomplish a natural epimorphism $Q_Y\otimes_C M\to M(Y)$ which should be injective.
Currently I think that the formula $$ M(Y)\to Q_Y\otimes_C M;\quad m\mapsto\mathrm{id}_Y\otimes m $$ gives indeed a (two-sided) inverse. Is this correct?