Given a set S with an equivalence relation ~, we can form a groupoid consisting of all ordered pairs of equivalent elements, i.e., all pairs $(s_1,s_2)$ where $s_1 \sim s_2$. The composition is then $(s_1,s_2) \circ (s_3,s_4) = (s_1,s_4)$ defined when $s_2 = s_3$.

It's such an obvious example of a groupoid that I'm embarrassed that I missed it. I'm sure it's in a textbook somewhere. I would have saved

*a lot*of time if I had found and read that textbook a week ago.

**Edit**

Oh! Look! There it is! It's on wikipedia. fml.

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