Almost ff-universality Implies Q-universality

A concrete category Q is finite-to-finite (algebraically) almost universal if the category of graphs and graph homomorphisms can be embedded into Q in such a way that finite Q-objects are assigned to finite graphs and non-constant Q-morphisms between any Q-objects assigned to graphs are exactly those arising from graph homomorphisms. A quasivariety Q of algebraic systems of a finite similarity type is Q-universal if the lattice of all subquasivarieties of any quasivariety R of algebraic systems of a finite similarity type is isomorphic to a quotient lattice of a sublattice of the subquasivariety lattice of Q. This paper shows that any finite-to-finite (algebraically) almost universal quasivariety Q of a finite type is Q-universal.