  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 $\mathrm{int}M$. 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 $\mathrm{int}M$ 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.
