Tensor product of A${}_{\mathrm{\infty }}$-categories

In this paper we define the tensor product of two A${}_{\mathrm{\infty }}$-categories and two A${}_{\mathrm{\infty }}$-functors. This tensor product makes the category of A${}_{\mathrm{\infty }}$-categories symmetric monoidal (up to homotopy), and the category A${}_{\mathrm{\infty }}$Cat${}^u$/${}_{\approx }$ a closed symmetric monoidal category. Moreover, we define the derived tensor product making Ho(A${}_{\mathrm{\infty }}$Cat), the homotopy category of the A${}_{\mathrm{\infty }}$-categories, a closed symmetric monoidal category. We provide also an explicit description of the internal homs in terms of A${}_{\mathrm{\infty }}$- functors.