Relating Quotient Completions via Categorical Logic

In previous work we introduced the notion of elementary quotient completion with respect to an elementary doctrine. We also generalized the notion of exact completion of a regular category as an exact completion of an existential elementary doctrine. Here we characterize when the elementary quotient completion of an elementary existential doctrine coincides with an exact completion. We do this by employing the categorical logic of the various notions of doctrines involved in our analysis. The outcome is that the two completions coincide when a choice rule holds in the starting existential elementary doctrine.