  We lift the standard equivalence between fibrations and indexed categories to
an equivalence between monoidal fibrations and monoidal indexed categories,
namely weak monoidal pseudofunctors to the 2-category of categories. In doing
so, we investigate the relation between this `global' monoidal structure where
the total category is monoidal and the fibration strictly preserves the
structure, and a `fibrewise' one where the fibres are monoidal and the
reindexing functors strongly preserve the structure, first hinted by Shulman.
In particular, when the domain is cocartesian monoidal, lax monoidal structures
on the functor to the 2-category of categories correspond to lifts of the
functor to the 2-category of monoidal categories. Finally, we give examples
where this correspondence appears, spanning from the fundamental and family
fibrations to network models and systems.
