Let $L\dashv R$ a Quillen pairs an adjunction between model categories $M,M'$ with $L$ preserving cofibrations and $R$ fibrations. From then we can construct an adjunction $\mathbb LL\dashv \mathbb RR$ between the total derived functors. Now suppose it's actually an equivalence. We want to show that $L\dashv R$ is an Quillen equivalence, that is for $x\in M$ cofibrant, $x'\in M'$ fibrant and $f\in M'(L(x),x')$ we have : $f\in WE'\iff f^{\#}\in WE$ ($f^{\#}$ is obtained via the adjunction).
My question is : how do we show this ?
Suppose for instance $f$ is a weak equivalence, we write $\eta :Id_M\to R\circ L$, $\epsilon:L\circ R\to Id_{M'}$ the natural transformations we get $f^{\#}=R(f)\circ \eta_x$ so by the 2-out-3 it's enough to show that $R(f),\eta_x\in WE$ but I don't go anywhere. I also don't know how to use the equivalence $\mathbb LL\dashv \mathbb RR$
See Proposition 1.3.13 in Hovey’s Model Category.