Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes:

In many cases, the category $\cal{M}od_A\left(\cal{C}\right)$ of $A$-modules in $\cal{C}$ inherits the structure of a symmetric monoidal category with respect to the relative tensor product over $A$.

Where can I find conditions, details and proofs of this (seemingly) elementary fact? (I didn't find it in the book I searched - MacLane, Barr & Wells, or in books about operads etc.)

I also need the facts that "extension of scalars" $-\otimes A$ is the left adjoint of the forgetful functor and it commutes with the tensor product. And the case when $\otimes$ distriute over the coproduct (or the bi-product).

I think I can prove most of these facts, but I want to be sure about them, and it would also be much easier to refer to them.

4more comments