  Liftable pairs of adjoint functors between braided monoidal categories in the
sense of \cite{GV-OnTheDuality} provide auto-adjunctions between the associated
categories of bialgebras. Motivated by finding interesting examples of such
pairs, we study general pre-rigid monoidal categories. Roughly speaking, these
are monoidal categories in which for every object $X$, an object $X^{\ast}$ and
a nicely behaving evaluation map from $X^{\ast}\otimes X$ to the unit object
exist. A prototypical example is the category of vector spaces over a field,
where $X^{\ast}$ is not a categorical dual if $X$ is not finite-dimensional. We
explore the connection with related notions such as right closedness, and
present meaningful examples. We also study the categorical frameworks for
Turaev's Hopf group-(co)algebras in the light of pre-rigidity and closedness,
filling some gaps in literature along the way. Finally, we show that braided
pre-rigid monoidal categories indeed provide an appropriate setting for
liftability in the sense of loc. cit. and we present an application, varying on
the theme of vector spaces, showing how -- in favorable cases -- the notion of
pre-rigidity allows to construct liftable pairs of adjoint functors when right
closedness of the category is not available.
