Why is Quillen equivalence used as the notion of equivalence of model categories?
So given two model categories $C,D$, $$L \dashv R:C \rightarrow D$$ is called a Quillen adjunction if $L$ preserves cofibrations and $R$ preserves fibrations (or some other equivalent formulations).
This induces a map in the homotopy categores: $$\Bbb L L \dashv \Bbb R R: Ho(C) \rightarrow Ho(D)$$
this is a Quillen equivalence if this adjunction is in fact an equivalence of categories.
What properties capture the idea that this is the right notion of "equivalence" by passing to homotopy category: rather than simply requiring adjoint equivalence of $C,D$?
References would help - I am aware of nlab post but am not satisfied.