It is shown that a Fréchet-Schwartz space $E$ with an absolute basis has the property $\overline{\overline{\Omega}}$ if and only if every holomorphic function on $D\times E$ with values in a pseudoconvex space having a Stein morphism into a complex Lie group, where $D$ is a Stein space, is of uniform type. In the scalar case, where $E$ is a nuclear Fréchet space, the result was established by Meise and Vogt [5].