  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 of the author 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.
