The tensor product of presentable -categories is the product in the symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories. See there for more details.
This is the -category which is ‘the universal recipient of a bilinear functor’ from . Here, we think of coproducts in and as addition, and then if a functor preserves colimits in each variable, in particular it preserves coproducts and so is ‘bilinear’. Such a bilinear functor will factor uniquely (in a homotopic sense) through a universal bilinear functor , just like for bilinear maps and tensor products of abelian groups.