Tensor product of $A_{\infty}$ algebra morphisms, reference

Let $\Bbbk$ be a field of charachteristic zero. Let $(A, m_{\bullet}^{A})$ be an unital $A_{\infty}$ algebra. Let $B$ be a differential graded algebra. Then $B\otimes A$ carries an $A_{\infty}$ algebra structures given by $$m_{n}(b_{1}\otimes a_{1}, \dots, b_{n}\otimes a_{n})=\pm (b_{1}\cdots b_{n})\otimes m_{n}^{A}(a_{1}, \dots, a_{n})$$ where the sign is given by the sign rule. For $i=1,2$ let $I_{i}$ be a differential graded algebra equipped with dg algebra maps $s^{i}_{j}\: : \:I_{i}\to \Bbbk$ for $j=0,1$ wich are quasi isomorphism. Let $f_{\bullet}\: : \: I_{1}\to I_{2}$ be an $A_{\infty}$ morphism between differential graded algebras wich is a quasi isomorphism and assume that $s^1_{j}=s^2_{j}f$ for $j=0,1$ . I wonder if it is possible to construct a quasi isomorphism of $A_{\infty}$ algebras $$(f\otimes 1_{A})_{\bullet}\: : \: I_{1}\otimes A\to I_{2}\otimes A$$ such that

$s^1_{j}\otimes 1_{A}=(s^2_{j}\otimes 1_{A})(f\otimes 1_{A})_{\bullet}$

Could you help me? I need a reference about the existence of such a map.

P.S:It seems to me that the existence is related with the fact that both the objects are "path objects" from the model category point of view.

• I think the formular you write will not give you an $A_\infty$ algebra structre, because you don't use $m^B_1$. I was wondering if the tensor product of a DGA and an $A_\infty$ algebra is still an $A_\infty$ algebra. Could you give a reference for this claim? – Hang Oct 29 '18 at 21:41