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.