  We define and study opfibrations of $V$-enriched categories when $V$ is an
extensive monoidal category whose unit is terminal and connected. This includes
sets, simplicial sets, categories, or any locally cartesian closed category
with disjoint coproducts and connected unit. We show that for an ordinary
category $B$, there is an equivalence of 2-categories between $V$-enriched
opfibrations over the free $V$-category on $B$, and pseudofunctors from $B$ to
the 2-category of $V$-categories. This generalizes the classical
($Set$-enriched) Grothendieck correspondence.
