We analyze quantum field theories on spacetimes M with timelike boundary from a model-independent perspective. We construct an adjunction which describes a universal extension to the whole spacetime M of theories defined only on the interior . The unit of this adjunction is a natural isomorphism, which implies that our universal extension satisfies Kay's F-locality property. Our main result is the following characterization theorem: Every quantum field theory on M that is additive from the interior (i.e., generated by observables localized in the interior) admits a presentation by a quantum field theory on the interior and an ideal of its universal extension that is trivial on the interior. We shall illustrate our constructions by applying them to the free Klein-Gordon field.