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In this work we fill the gap between the elementary quotient completion ofMaietti and Rosolini and the exact completion of a weakly left exact categoryof Carboni and Vitale. In order to do that, we generalize Lawvere's elementarydoctrines for categories with weak finite products and call such structuresbiased elementary doctrines. We provide two main constructions. The first oneis called strictification, which produces an elementary doctrine from a biasedone. The second is an extension of the elementary quotient completion, whichgeneralizes the exact completion of a weakly left exact category even in caseof weak finite products.

Biased elementary doctrines and quotient completions