  In a previous paper, the author and his collaborators studied the phenomenon
of isotropy in the context of single-sorted equational theories, and showed
that the isotropy group of the category of models of any such theory encodes a
notion of inner automorphism for the theory.
  Using results from the treatment of combination problems in term rewriting
theory, we show in this article that if $\mathbb{T}_1$ and $\mathbb{T}_2$ are
(disjoint) equational theories satisfying minimal assumptions, then any free,
finitely generated model of the disjoint union theory $\mathbb{T}_1 +
\mathbb{T}_2$ has trivial isotropy group, and hence the only inner
automorphisms of such models, i.e. the only automorphisms of such models that
are coherently extendible, are the identity automorphisms.
  As a corollary, we show that the global isotropy group of the category of
models $(\mathbb{T}_1 + \mathbb{T}_2)\mathsf{mod}$, i.e. the group of
invertible elements of the centre of this category, is the trivial group.
