  Let A be a k-algebra over a commutative ring k. By the renowned Tannaka-Krein
reconstruction, liftings of the monoidal structure from k-modules to A-modules
correspond to bialgebra structures on A and liftings of the closed monoidal
structure correspond to Hopf algebra structures on A. In this paper, we
determine conditions on A that correspond to liftings of the closed structure
alone, i.e. without considering the monoidal one, which lead to the notion of
what we call a gabi-algebra. First, we tackle the question from the general
perspective of monads, then we focus on the set-theoretic and the linear
setting. Our main and most surprising result is that a normal gabi-algebra,
that is an algebra A whose category of modules is (associative and unital
normal) closed with closed forgetful functor to k-modules, is automatically a
Hopf algebra (thus justifying our title).
