$\{M_i\}_{i\in I}$ is a directed system with direct limit $M$. For each $i\in I$, $N_i\subseteq M_i$ is a submodule and $\{N_i\}_{i\in I}$ with the restriction maps is also a directed system with direct limit $N$. So there is a natural map from $N$ to $M$.
Is this map always injective? If the answer is no, what is the condition for the map to be injective?
This problem is not obvious to me. Consider the example: choose $x$ and $y$ from $M_i$, such that $x+y\in N_i$, $\mu_{i,j}(x)\in N_j$ and $\mu_{i,k}(y)\in N_k$ for some $j, k\ge i$ (assume they exist). Then obviously $$ x-\mu_{i,j}(x)+y-\mu_{i,k}(y)=0\in M. $$ Now how to prove that $x-\mu_{i,j}(x)+y-\mu_{i,k}(y)=0$ in $N$? Or such $x$ and $y$ do not exist at all?
As a visitor who was wondering about the same question, here is my answer.
Let $\textbf{M} = ((M_i),(\mu_{ij}))$ be a direct system with direct limit $(M, (\mu_i))$. Suppose that for each $i$, there are sub-modules $N_i \le M_i$ such that $\mu_{ij}(N_i) \subseteq N_j$. Then, it is straightforward to verify that $ \textbf{N} =((N_i), (\mu_{ij} \vert_{N_i}))$ is also a direct system; let the direct limit be $(N, (\nu_i))$.
Rather than working with the troublesome construction of the direct limit, consider the family of inclusion homomorphisms $\iota_i : N_i \to M_i$. It is clear that $\iota_j \circ \mu_{ij} \vert_{N_i} = \mu_{ij} \circ\iota_i$, so this gives a homomorphism of direct systems $\textbf{M} \to \textbf{N}$, which in turn descends to a homomorphism $\iota : M \to N$ satisfying $\iota \circ \nu_i = \mu_i \circ \iota_i$.
We claim that $\iota$ is injective. Let $n \in N$. Then, $\exists i$ s.t. $\nu_i(n_i) = n$.
Suppose that $\iota(n)=0$.
$$\implies \iota(\nu_i(n_i)) = 0$$ $$\implies (\mu_i \circ \iota_i)(n_i) = 0$$ $$\implies \mu_i(n_i) = 0$$ $$\implies \exists j \; \text{ s.t. } \mu_{ij} (n_i) = 0$$ $$\implies \exists j \; \text{ s.t. } \mu_{ij} \vert_{N_i} (n_i) = 0$$
$$\implies \nu_i (n_i) = (\nu_j \circ \mu_{ij} \vert_{N_i} )(n_i) = 0$$
$$\implies n = 0$$
Hence, $\iota$ is injective, and $N$ can be identified as a sub-module of $M$.