Quillen equivalence for under-categories

If $c$ is cofibrant and $d$ is fibrant, then $c \to R(d)$ is a weak equivalence if and only if $L(c) \to d$ is. Now, we can prove that if $L(c) \to d$ is a weak equivalence, $c$ is cofibrant, and either $d$ is cofibrant or $D$ is left proper, then $\hat{L} \dashv \hat{R}$ is a Quillen equivalence. First, note that $\hat{L} \dashv \hat{R}$ is always a Quillen adjunction. Let us prove that it is a Quillen equivalence under these assumptions. Since $R$ reflects weak equivalences between fibrant objects, so does $\hat{R}$. Thus, we just need to prove that the map $c_1 \to RL(c_1) \to RSL(c_1) \to RS(d_1)$ is a weak equivalence for every cofibration $i : c \to c_1$, where $S$ is a fibrant replacement functor and $L(c_1) \to d_1$ is the pushout of $f : L(c) \to d$ along $L(i) : L(c) \to L(c_1)$. The composition of the first two maps is a weak equivalence since $L \dashv R$ is a Quillen equivalence and $c_1$ is cofibrant. To prove that the last map is a weak equivalence, note that $S$ preserves weak equivalences and $R$ preserves weak equivalences between fibrant objects. Thus, we just need to prove that $L(c_1) \to d_1$ is a weak equivalence. Since it is a pushout of a weak equivlence along a cofibration, this follows from our assumptions.

Actually a while ago Benoit Fresse gave me an answer to this question. I figured I should post it here.

The condition on $C$ and $D$ is that they should be left proper and $R$ should preserve weak equivalences.