Lemma 10.131.5. Let $I$ be a directed set. Let $(R_ i \to S_ i, \varphi _{ii'})$ be a system of ring maps over $I$, see Categories, Section 4.21. Then we have

where $R \to S = \mathop{\mathrm{colim}}\nolimits (R_ i \to S_ i)$.

Lemma 10.131.5. Let $I$ be a directed set. Let $(R_ i \to S_ i, \varphi _{ii'})$ be a system of ring maps over $I$, see Categories, Section 4.21. Then we have

\[ \Omega _{S/R} = \mathop{\mathrm{colim}}\nolimits _ i \Omega _{S_ i/R_ i}. \]

where $R \to S = \mathop{\mathrm{colim}}\nolimits (R_ i \to S_ i)$.

**Proof.**
This is clear from the defining presentation of $\Omega _{S/R}$ and the functoriality of this described above.
$\square$

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