Let $A^{\bullet}$, $B^{\bullet}$ be two differential graded algebras (dga) and $f: A^{\bullet}\to B^{\bullet}$ be a differential graded homomorphism between them. Now let $R$ be another algebra ($R$ can be considered as a dga concentrated in degree $0$.) and $\theta: R\to A^{\bullet}$ be a homomorphism of dgas. The composition with $f$ gives a homomorphism $f\circ \theta:R\to B^{\bullet}$．
Now we can consider the centralizer of $R$ in $A^{\bullet}$ and $B^{\bullet}$. In more details we define $$ A^{\bullet,R}=\{a\in A^{\bullet}|a\theta(r)=\theta(r)a=0, \forall r\in R\} $$ and similarly we can define $B^{\bullet,R}$. It is clear that $f$ induces a homomorphism $f^R: A^{\bullet,R}\to B^{\bullet,R}$.
Now I think that even if $f: A^{\bullet}\to B^{\bullet}$ is a quasi-isomorphism, the induced homorphism $f^R: A^{\bullet,R}\to B^{\bullet,R}$ is not necessarily a quasi-isomorphism. However I cannot find a counter-example. Is there a counter example for this claim?