I believe the following statement is true under certain technical conditions. 


Let $$L\colon C \rightleftharpoons D\colon R$$ be a Quillen equivalence. Suppose we are given objects $c\in C$, $d\in D$, and an equivalence $c\to R(d)$. (Perhaps we need to assume that $d$ is fibrant, but $c$ doesn't have to be cofibrant.) Then one has a Quillen equivalence of under-categories: $\hat L\colon c/C\rightleftharpoons d/D\colon \hat R$, where $\hat R$ is an obvious functor sending $d\to d\_1$ to the composition $c\to R(d)\to R(d\_1)$; and $\hat L$ sends $c\to c\_1$ to 
$$
d\to \mathrm{colim}\bigl(d\leftarrow L(c)\rightarrow L(c\_1)\bigr).
$$


I wonder what are the conditions under which this is true? Any reference? In fact I am interested in the case C and D are categories of bimodules over operads.



Quillen equivalence for under-categories
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. 


