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It is known that there exists a natural functor $\Phi$ from Lie supergroupsto super Harish-Chandra pairs. A functor going backwards, that associates a Liesupergroup with each super Harish-Chandra pair, yielding an equivalence ofcategories, was found by Koszul [18]; this result was later extended by otherauthors, to different levels of generality, but always elaborating on Koszul'soriginal idea.  In this paper, I provide two new backwards equivalences, i.e. two differentfunctors $\Psi^\circ$ and $\Psi^e$ that construct a Lie supergroup (thought ofas a special group-valued functor) out of a given super Harish-Chandra pair, sothat any Lie supergroup is recovered from its naturally associated superHarish-Chandra pair; more precisely, both $\Psi^\circ$ and $\Psi^e$ arequasi-inverse to the functor $\Phi$.

Lie supergroups vs. super Harish-Chandra pairs: a new equivalence