  We show that each rigid monoidal category A over a field defines a family of
universal tensor categories, which together classify all faithful monoidal
functors from A to tensor categories. Each of the universal tensor categories
classifies monoidal functors of a given 'homological kernel' and can be
realised as a sheaf category, not necessarily on A. This yields a theory of
'local abelian envelope' which completes the notion of monoidal abelian
envelopes.
