I know that: If $f$ is a least function for $\mathcal{U}\subset\mathcal{P}(\kappa)$ then: $\mathcal{U}$ is $(\mu,\kappa)$-regular iff $\{\xi<\kappa|cf(f(\xi))<\mu\}\in\mathcal{U}$. With that I want to prove that any such $\mathcal{U}$ with a least function cannot be $(\omega,\kappa)$-regular. (Here least function for $\mathcal{U}$ is an unbounded function such that $\{\xi|g(\xi)<f(\xi)\}\in\mathcal{U}$ implies boundedness for $g$.) Also I know $\kappa$ is regular and $\mathcal{U}$ is uniform.
If $\{\xi<\kappa|cf(f(\xi))<\omega\}\in\mathcal{U}$ is to say that $f$ takes 'mostly' successor values which does not seems weird to me. ObviouslyIi am missing something.