A remark on the Galois-type correspondence between ideal coideals and comodule subrings of a Hopf algebroid

We exhibit a bijective correspondence between certain left ideal coideals in a Hopf algebroid for which the resulting quotient is a coequalizer and certain right coideal subrings which are themselves an equalizer, remarkably improving a recent result obtained in collaboration with L. El Kaoutit, A. Ghobadi, and J. Vercruysse. Interpreting this result in the Hopf algebra setting provides a bijective correspondence which extends the previously known cases.