Freely add all quotients to a category I would like to know if there is some uniform construction out of a given category $\mathcal C$ that freely throws in all quotients,to form a new category $\mathcal C'$. Preferably $\mathcal C'$ has all (small?) quotients, but if it only contains the quotients out of $\mathcal C$ I can also live with it.
I know that the presheaf category $[\mathcal C^{\mathrm {op}} , \mathrm{Set}]$ is the free cocompletion of $\mathcal C$, which means that I get all the quotients, but I also get a whole bunch of other stuff. Maybe it has to do with $\mathrm{Set}$ contains all the colimits. In fact $\mathrm{Set} = [\mathbb 1^{\mathrm {op}}, \mathrm{Set}]$, which is the free cocompletion of the category $\mathbb 1$. so maybe we should consider $[\mathcal C^{\mathrm {op}}, \mathbb Q]$ where $\mathbb Q$ is something like the category that somehow contains all quotients generated from $\mathbb 1$? But then $\mathbb Q = \mathbb 1$ since there are no new quotients to add at all...
On the other hand, maybe this is related to some sort of categorified process of "setoid-ification".
Is there existing results concerning this question? Any help is appreciated.
 A: In general, the way to construct a free completion of a category under only some colimits is to take a full subcategory of the presheaf category $[C^{\rm op},\rm Set]$ that's the closure of the representables under the colimits in question.  In particular we can do this for quotients, although there are different meanings we might pick for "quotient".

*

*If by "quotient" we mean simply a coequalizer, then the closure of the representables under coequalizers is the free cocompletion under coequalizers, with a universal property relative to mapping into other categories with coequalizers.


*If by "quotient" we mean the quotient of an internal equivalence relation, then the closure of the representables under such quotients is called the ex/lex completion.  As long as the category $C$ already has finite limits, so that it makes sense to talk about internal equivalence relations therein, this has a universal property relative to finite-limit-preserving functors into exact categories.  This can be generalized to other kinds of exact completions.
