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

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