  Given an operad $\mathcal{O}$, we define a notion of weak
$\mathcal{O}$-monoids -- which we term $\mathcal{O}$-pseudomonoids -- in a
2-category. In the special case with the 2-category in question is the
2-category $\mathsf{Cat}$ of categories, this yields a notion of
$\mathcal{O}$-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
$\mathcal{O}$-monoids in the 2-categories of discrete fibrations and of
category-indexed sets. Using the classical Grothendieck construction, we
thereby obtain an $\mathcal{O}$-monoidal Grothendieck construction relating lax
$\mathcal{O}$-monoidal functors into Set to strict $\mathcal{O}$-monoidal
functors which are also discrete fibrations.
