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There is a reflective coequalizer completion, much like the regular and exact completions. The construction seems to be due to Pitts, see e.g. Ch17 of the book on Algebraic Theories. What would be the right way to add this to the nlab? There are probably more references, and the page on regular and exact completions could be improved a little. What is the nPOV?
You could put it at reflexive coequalizer and link to it from regular and exact completions.
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