Lemma 10.8.2. Let $(M_ i, \mu _{ij})$ be a system of $R$-modules over the preordered set $I$. The colimit of the system $(M_ i, \mu _{ij})$ is the quotient $R$-module $(\bigoplus _{i\in I} M_ i) /Q$ where $Q$ is the $R$-submodule generated by all elements

\[ \iota _ i(x_ i) - \iota _ j(\mu _{ij}(x_ i)) \]

where $\iota _ i : M_ i \to \bigoplus _{i\in I} M_ i$ is the natural inclusion. We denote the colimit $M = \mathop{\mathrm{colim}}\nolimits _ i M_ i$. We denote $\pi : \bigoplus _{i\in I} M_ i \to M$ the projection map and $\phi _ i = \pi \circ \iota _ i : M_ i \to M$.

**Proof.**
This lemma is a special case of Categories, Lemma 4.14.12 but we will also prove it directly in this case. Namely, note that $\phi _ i = \phi _ j\circ \mu _{ij}$ in the above construction. To show the pair $(M, \phi _ i)$ is the colimit we have to show it satisfies the universal property: for any other such pair $(Y, \psi _ i)$ with $\psi _ i : M_ i \to Y$, $\psi _ i = \psi _ j\circ \mu _{ij}$, there is a unique $R$-module homomorphism $g : M \to Y$ such that the following diagram commutes:

\[ \xymatrix{ M_ i \ar[rr]^{\mu _{ij}} \ar[dr]^{\phi _ i} \ar[ddr]_{\psi _ i} & & M_ j\ar[dl]_{\phi _ j} \ar[ddl]^{\psi _ j} \\ & M \ar[d]^{g}\\ & Y } \]

And this is clear because we can define $g$ by taking the map $\psi _ i$ on the summand $M_ i$ in the direct sum $\bigoplus M_ i$.
$\square$

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