An O-monoidal Grothendieck construction

Given an operad O, we define a notion of weak O-monoids -- which we term O-pseudomonoids -- in a 2-category. In the special case with the 2-category in question is the 2-category Cat of categories, this yields a notion of O-monoidal category, which in the case of the associative and commutative operads retrieves unbiased notions of monoidal and symmetric monoidal categories, respectively. We carefully unpack the definition of O-monoids in the 2-categories of discrete fibrations and of category-indexed sets. Using the classical Grothendieck construction, we thereby obtain an O-monoidal Grothendieck construction relating lax O-monoidal functors into Set to strict O-monoidal functors which are also discrete fibrations.