Tensor structures on fibered categories

Let $\mathcal{S}$ be a small category admitting binary products. We show that the whole theory of monoidal $\mathcal{S}$-fibered categories can be rephrased using the external tensor product everywhere in place of the usual internal tensor product. More precisely, we construct a canonical dictionary relating the classical structures and properties of the internal tensor product with analogous structures and properties of the external tensor product: this applies to associativity, commutativity and unit constraints, to projection formulae, as well as to monoidality of morphisms between monoidal $\mathcal{S}$-fibered categories.
Comment: 78 pages