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I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.
The two categories C and D had got mixed up at one place, so $\eta_d:d\to R(L(d))$ was there instead of $\eta_c:c\to R(L(c))$. I think it is right now. (I checked in the reference given and they seem to have the mistake as well. Confusing.)
In fact this part (proposition 2.3) of the entry seems to be in bit of a mess, as $R$ and $L$ or $C$ and $D$ have somehow ended up being swapped from earlier on in the entry. I am confused!!!
The given sufficient condition for $C/S \leftrightarrows C/T$ to be a Quillen equivalence is actually a necessary condition too.
Then let’s make this part of the statement, not hide it in the proof.
I have reworked both the statement and the proof a fair bit (here), adding more explanation throughout (the key use of 2-out-of-3 wasn’t even mentioned before).
Also I cross-linked with base change Quillen adjunction. The whole proposition would rather be found there than here, and so I am copying it over now.
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