Does every covariant functor $F : ~_R\mathcal{M} \rightarrow \mathcal{C}$ on module category $_R\mathcal{M}$ preserve inclusion? I have proceeded in the following way.
Suppose $A \subseteq B$ in $_R\mathcal{M}$ then the natural inclusion map $i$ acts as an identity $1_A$ on $A$, then $F(i)$ must act as an identity on $F(A)$, therefore $F(A) \subseteq F(B)$. But clearly the functor $\mathbb{Z}_n \otimes_\mathbb{Z} -$ doesnot preserve the inclusion $\mathbb{Z} \subseteq \mathbb{Q}$. Am I overlooking something? Any help would be appreciated.