Functors between C*-relations

The paper deals with functors acting between categories which arise in the theory of universal $C^*$-algebras. In the framework of a categorical approach to the notion of a universal $C^*$-algebra generated by a set of generators subject to relations, T.A.~Loring introduced and studied categories called the $C^*$-relations. Given a set $X$, a $C^*$-relation on $X$ is a category whose objects are functions from $X$ to $C^*$-algebras and morphisms are $\ast$-homomorphisms of $C^*$-algebras making the appropriate triangle diagrams commute. Moreover, these functions and $\ast$-homomorphisms satisfy certain natural axioms. A $C^*$-relation is said to be compact if it determines a universal $C^*$-algebra. In this paper, it is shown that every functor between arbitrary compact $C^*$-relations is a functor between $\ast$-polynomial relations on the same set $X$ up to isomorphisms of categories. Using ideals in the algebras of involutive polynomials, we construct the factorization functors which constitute an important class of functors between $C^*$-relations. To study properties of the factorization functors, we introduce the notion of a soft image of a functor and establish a criterion for an object to belong to a soft image.