Hubbard-corrected density-functional theory (DFT+U) is a popular tool for first-principles modeling of materials with localized d or f electrons, but its on-site corrections tend to over-localize charge and break covalent bonds. Inter-site +V corrections were introduced to counter this and are now widely used, but a formal justification has been lacking. Here we show that — to first order in V/U — inter-site corrections are exactly equivalent to on-site DFT+U evaluated on a density-matrix-dependent redefinition of the Hubbard projectors, hybridized with those of neighboring sites, providing insight into the explicit mechanism by which V affects covalency. If the projectors are held frozen, as is common practice, the equivalence partially breaks down. Beyond reinterpreting the formalism, these results sharpen the questions of how V should be computed and what the Hubbard subspaces fundamentally are.