Suppose we have the ultrafilter definition of a supercompact cardinal. Now, we know if $\kappa$ is supercompact and indestructible by any <$\kappa$ directed closed forcing poset in V then $\kappa$ remains supercompact in V[G] after forcing with a <$\kappa$ directed closed poset.
But then can we say for all $\lambda\geq\kappa$, the corresponding $\kappa$ complete normal ultrafilter $\mathcal{U}_1$ on $\mathcal{P}_{\kappa}(\lambda)$ in V is the same as the corresponding $\kappa$ complete normal ultrafilter $\mathcal{U}_2$ on $\mathcal{P}_{\kappa}(\lambda)$ in V[G] ? If not, can we generate $\mathcal{U}_2$ from $\mathcal{U}_1$ in some way, or is there any type of connection between $\mathcal{U}_2$ and $\mathcal{U}_1$ ?