Categorical criterion for existence of universal $C^*$–algebras

We deal with categories, which determine universal $C^*$–algebras. These categories are called the compact $C^*$–relations. They were introduced by T.A. Loring. Given a set $X,$ a compact $C^*$–relation on $X$ is a category, the objects of which 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 axioms. In this article, we prove that every compact $C^*$–relation is both complete and cocomplete. As an application of the completeness of compact $C^*$–relations, we obtain the criterion for the existence of universal $C^*$–algebras.