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Is the module of Kähler differentials a coend?

Let $\phi\colon R\to S$ be a ring map. The module of Kähler differentials $\Omega_{S/R}$ of $\phi$ can be constructed as the following coequaliser: $$\left(\bigoplus_{(a, b)\in S^2} S[(a, b)]\right) \...
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