question on adjoint lifting

question on adjoint lifting

[Adriana Balan]

Dear categorists,

Could you help me by pointing out a reference for the following situation:
consider two categories C and D and monads H:C-->C and K:D-->D. Let
T:C-->D be a functor having left and right adjoint non-isomorphic. Assume
that exists a lifting of T to the Eilenberg-Moore categories of algebras
T^:Alg(M)-->Alg(N), corresponding to a distributive law lambda: KT=>TH.
Suppose Alg(H) has reflexive coequalizers and lambda is an isomorphism.
Then according to Johnstone, Adjoint lifting theorems for categories of
algebras, T^ has both left and right adjoints.  My question is the
following: under which conditions the left and right adjoints of T^ are
isomorphic (i. e. T^ is a Frobenius functor)? Has anyone already
considered this situation? This is my first time using this mailing list,
and I would appreciate any help.

Thanks,
Adriana Balan


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