A new equivalence between super Harish-Chandra pairs and Lie supergroups

It is known that there exists a natural functor Phi from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul (1983), and later generalized by several authors. We provide two new backwards equivalences, i.e., two different functors Psi^circ and Psi^e that construct a Lie supergroup (thought of as a special group-valued functor) out of a given super Harish-Chandra pair, so that both Psi^circ and Psi^e are quasi-inverse to the functor Phi.