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    $\begingroup$ By well-powered do you mean "connected to a continuous energy source"? $\endgroup$ Commented Feb 13, 2021 at 12:45
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    $\begingroup$ @AsafKaragila Exactly. Unfortuately, "co-well-powered" means that it's also connected to a continuous energy drain. So the above theorem that $(1) \Leftrightarrow (2)$ is really a weak form of the 0th law of thermodynamics. $\endgroup$ Commented Feb 13, 2021 at 12:51
  • $\begingroup$ One way to think about this could be to study the presheaf $\mathsf{RSub}$ of regular subobject and the copresheaf $\mathsf{RQuo}$ of regular quotient. Now one could try to say that there must be an injective map $\mathsf{RQuo} \to 2^\mathsf{RSub}$ (notice that the variance now match). Similar tricks are used by Freyd in Sec 5 of "On the concreteness of certain categories", for very different purposes. $\endgroup$ Commented Feb 13, 2021 at 12:59
  • $\begingroup$ @IvanDiLiberti That sounds promising. The usual ways I know of relating subobjects and quotients -- taking kernels / cokernels, taking kernel pairs / quotienting by a congruence etc. seem to require stronger exactness properties than I'm assuming here. It would be interesting if there were something like this which didn't require any sort of exactness. $\endgroup$ Commented Feb 13, 2021 at 13:00
  • $\begingroup$ That was my idea. It's hard to say without spending time on a whiteboard. $\endgroup$ Commented Feb 13, 2021 at 13:01