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We show that supercompactness and strong compactness can be equivalent evenas properties of pairs of regular cardinals. Specifically, we show that if Vmodels ZFC + GCH is a given model (which in interesting cases containsinstances of supercompactness), then there is some cardinal and cofinalitypreserving generic extension V[G] models ZFC + GCH in which, (a)(preservation) for kappa <= lambda regular, if V models ``kappa is lambdasupercompact'', then V[G] models ``kappa is lambda supercompact'' and so that,(b) (equivalence) for kappa <= lambda regular, V[G] models ``kappa is lambdastrongly compact'' iff V[G] models ``kappa is lambda supercompact'', exceptpossibly if kappa is a measurable limit of cardinals which are lambdasupercompact.

On the strong equality between supercompactness and strong compactness