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Exercise 4.8.7 in Vol. 2 of Borceux's Handbook, asks the following:
Let and be monads on , and be a morphism of monads, which induces a functor over . Show that is full and faithful iff is epic for all in .
That being objectwise epic implies that is full and faithful follows from a simple diagram chase. I am completely stumped by the other direction though. Does anyone have any hints or ideas?
I'll say briefly what I've tried. Since is always faithful, the hypothesis says that if and are -algebras, and is an -algebra morphism , then is also an a -algebra morphism . To apply this hypothesis, it seems like I need to get my hands on two -algebras, and the only -algebras that are guaranteed to exist are the free ones. But such an argument seems like it will only be able to say something about the components of , and not those of .
I believe the statement in Borceux is incorrect: being pointwise epic is sufficient, but not necessary. The question of the induced functor being fully faithful is the topic of Karazeris–Velebil's Dense morphisms of monads. See Remark 3.5 there in particular.