A quasivariety of algebraic systems \(K\) is universal if every category of finite type (or, equivalently, the category \(G\) of all directed graphs) is isomorphic to a full subcategory of \(K\). If an embedding of \(G\) may be effected by a functor \(F: G\rightarrow K\) which assigns a finite algebraic system to each finite graph, then \(K\) is said to be finite-to-finite universal. Many familiar (quasi)varieties of algebraic systems are known to be finite-to-finite universal. For a quasivariety \(K\), let \(L(K)\) denote the lattice of all quasivarieties contained in \(K\). A quasivariety \(K\) of algebraic systems of finite type is \(Q\)-universal if for any quasivariety \(M\) of finite type, \(L(M)\) is a homomorphic image of a sublattice of \(L(K)\). In the paper under review the following main theorem is proved: If \(K\) is finite-to-finite universal quasivariety of algebraic systems of finite type, then \(K\) is \(Q\)-universal. Using this theorem, a number of quasivarieties are shown to be \(Q\)-universal.