Let $A$ and $B$ be two dg-algebras over a field $k$. Let $f, g: A\to B$ be two maps between dg-algebras. We call $f$ and $g$ chain homotopic if there exists a degree $-1$ map $h: A\to B$ such that $f-g=d_Bh+hd_A$.

Now let $M$ be a right $A$ dg-module. For a given map $f: A\to B$, we could obtain a right $B$ dg-module by tensor product: $M\mapsto M\otimes_AB$. To avoid confusion we denote it by $M\otimes^f_AB$.

My question is: if $f, g: A\to B$ are chain homotopic, then is it true that $M\otimes^f_AB$ and $M\otimes^g_AB$ are quasi-isomorphic or further isomorphic up to homotopy? If not, is there any counter-example?