  Given a cartesian closed category $\mathcal{V}$, we introduce an internal
category of elements $\int_\mathcal{C} F$ associated to a $\mathcal{V}$-functor
$F\colon \mathcal{C}^{\mathrm{op}}\to \mathcal{V}$. When $\mathcal{V}$ is
extensive, we show that this internal Grothendieck construction gives an
equivalence of categories between $\mathcal{V}$-functors
$\mathcal{C}^{\mathrm{op}}\to \mathcal{V}$ and internal discrete fibrations
over $\mathcal{C}$, which can be promoted to an equivalence of
$\mathcal{V}$-categories.
  Using this construction, we prove a representation theorem for
$\mathcal{V}$-categories, stating that a $\mathcal{V}$-functor $F\colon
\mathcal{C}^{\mathrm{op}}\to \mathcal{V}$ is $\mathcal{V}$-representable if and
only if its internal category of elements $\int_\mathcal{C} F$ has an internal
terminal object. We further obtain a characterization formulated completely in
terms of $\mathcal{V}$-categories using shifted $\mathcal{V}$-categories of
elements. Moreover, in the presence of $\mathcal{V}$-tensors, we show that it
is enough to consider $\mathcal{V}$-terminal objects in the underlying
$\mathcal{V}$-category $\mathrm{Und}\int_\mathcal{C} F$ to test the
representability of a $\mathcal{V}$-functor $F$. We apply these results to the
study of weighted $\mathcal{V}$-limits, and also obtain a novel result
describing weighted $\mathcal{V}$-limits as certain conical internal limits.
