A characterization of the category of a quasiprimitive class of universal algebras and its correspondences

If $\Omega$ is the class of all universal algebras with the system of operations $\Omega$, then all homomorphisms of $\Omega$-algebras form a category. In this article we find necessary and sufficient conditions under which an arbitrary category is isomorphic to a full subcategory of the category of $\Omega$-algebras closed with respect to direct products and subalgebras. We also find necessary and sufficient conditions under which a given category with involution is isomorphic to some full subcategory of the category of correspondences of $\Omega$-algebras.
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