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.



Freely add all quotients to a category
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".


1. 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.
2. If by "quotient" we mean the quotient of an [internal equivalence relation](https://ncatlab.org/nlab/show/congruence), then the closure of the representables under such quotients is called the [ex/lex completion](https://ncatlab.org/nlab/show/regular+and+exact+completions). 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.


