Direct limit functor on modules

Question

Let $(M_i,f_{i,j})_{i \in \mathbb{N}}$ and $(N_i,g_{i,j})_{i \in \mathbb{N}}$ be two directed system of modules over a ring $A$. Suppose $r:\varinjlim M_i \rightarrow \varinjlim N_i$ be a homomorphism. Is it true that it $r$ always arises from a compatible system of maps $r_i:M_i \rightarrow N_i$? i.e., is $r=\varinjlim r_i$ for some $r_i:M_i \rightarrow N_i$ or possibly for some $r_i:M_i \rightarrow N_{i+k}$ for some fixed $k$? If not, then under what assumptions it will be true? Any reference?

Answer 1

Take all $M_i=\mathbb Z_{(2)}$, the localization at $2$, with identity maps, to that the domain of $r$ is just $\mathbb Z_{(2)}$, and take the $N_i$ to be the subgroups $\mathbb Z\tfrac{1}{2^n}$ of $\mathbb Z_{(2)}$, with the maps the inclusions, so that the limit of the $M_i$ is also $\mathbb Z_{(2)}$. As $r$ take the identity map. Notice there are not many maps from the $M_i$ to the $N_j$.