  We investigate the concept of definable, or inner, automorphism in the
logical setting of partial Horn theories. The central technical result extends
a syntactical characterization of the group of such automorphisms (called the
covariant isotropy group) associated with an algebraic theory to the wider
class of quasi-equational theories. We apply this characterization to prove
that the isotropy group of a strict monoidal category is precisely its Picard
group of invertible objects. Furthermore, we obtain an explicit description of
the covariant isotropy group of a presheaf category.
