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.
Categories: math.CT