Although this looks like elementary, I have trouble understanding the proof of Theorem 3.1 at page 7 of this paper by Dominic Orchard (Univ. of Cambridge). As hypotheses we are given a comonad $D$, a monad $T$ and an adjunction $D \dashv T$. But then in the course of the proof (at the top of page 8), the author constructs the monad $T$ from the comonad $D$. Why is the constructed monad identical to the one in hypotheses?
from Hot Weekly Questions - Mathematics Stack Exchange
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