Definition: Let $(I,\leq)$ be a directed set. Let $(\{M_i\},\{f_{ij}\})$ be a direct system of $R$-modules. We define the injective limit to be the disjoint union of the $M_i$ modulo a certain equivalence relation: $$ \varinjlim M_i = \bigsqcup_i M_i \bigg/ {\sim}. $$ Here if $x_i \in M_i$ and $x_j \in M_j$ then $x_i \sim x_j$ if and only if there is some index $k \in I$ with $i\leq k$ and $j \leq k$ such that $f_{ik}(x_i) = f_{jk}(x_j)$. Intuitively, two elements of the disjoint union become equivalent if and only if they ``eventually become equal'' in the direct system. In proper notation, $(i,\alpha) \sim (j,\beta)$ if there is some $k$ larger than both $i,j$ for which $f_{ik}(\alpha) = f_{jk}(\beta)$. In particular, $(i,\alpha) \sim (k,f_{ik}(\alpha)) = (k,f_{jk}(\beta))\sim(j,\beta)$.
I'm trying to show that this direct limit recovers another definition of the direct limit in the category of $R$-modules, namely:
Definition: A direct limit $C$ is an object with morphisms $\alpha_i:C_i \rightarrow C$ such that for all $i \leq j$ we have $\alpha_i = \alpha_j f_{ij}$, and if $D$ is an object with arrows $\beta_i:C_i \rightarrow D$ for which $\beta_i = \beta_jf_{ij}$, then there is a unique arrow $\varphi:C \rightarrow D$ for which $\beta_i = \varphi\alpha_i$ and $\beta_j = \varphi\alpha_j$.
What I would like to do is construct maps $\alpha_i:M_i \rightarrow \sqcup_i M_i/{\sim}$ by composing natural injections $\varphi_i:M_i \rightarrow \sqcup_i M_i$ and projection $\pi:\sqcup_i M_i \rightarrow \sqcup_i M_i/{\sim}$.
However, I'm running into some problems. Can $\sqcup_i M_i$ be given an $R$-module structure so that the $\varphi_i$ are $R$-module homomorphisms? The $R$-action seems fine because you can have $r(i,m) = (i,rm)$, but there's no immediately obvious (to me) way to define $(i,m)+(j,n)$.
For that matter, what is the $R$-module structure on $\sqcup_i M_i/{\sim}$? Because these elements are equivalence classes so if I write $r[(i,m)] = [(i,rm)]$ I would need to show this is well-defined first? What would addition look like here? Thanks in advance for the clarification.