I present a new method to associate a Lie supergroup with a super Harish-Chandra pairs (=sHCp's), which provides an equivalence of categories between sHCp's and Lie supergroups. Namely, I provide a (new) functorial construction that, with each (real or complex) super Harish-Chandra pair, associates a (real or complex) Lie supergroup: this functor is then proved to be a quasi-inverse to the natural functor from Lie supergroups (up to details) to super Harish-Chandra pairs, so the two yield equivalences between the corresponding categories. The existence of such equivalences was known (possibly in different contexts, such as the smooth or the complex analytic one), but the construction I present is actually new - I present a different quasi-inverse functor - as it follows a totally different, more geometrical method