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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