Does universal enveloped C*-algebra is continuous functor?

Question

Let $K$ is category of *-algebras that have next property: for each $x \in B$ (where $B$ is *-algebra) $\sup_{\pi - bounded}||\pi(x)|| < \infty$ where $\pi : B \to B(H)$ - is some bounded representation in separable Hilbert space. And let $F : X \mapsto C^*(X)$ functor from category K to category of $C^*$-algebras, is that true that $F$ is continuous functor (commuting with inductive limits)?