  We introduce a natural generalization of the definition of a symmetric Hopf
algebroid, internal to any symmetric monoidal category with coequalizers that
commute with the monoidal product. Motivation for this is the study of
Heisenberg doubles of countably dimensional Hopf algebras $A$ as internal Hopf
algebroids over a (noncommutative) base $A$ in the category
$\mathrm{indproVect}$ of filtered cofiltered vector spaces introduced by the
author. One example of such Heisenberg double is internal Hopf algebroid
$U(\mathfrak{g}) \sharp U(\mathfrak{g})^*$ over universal enveloping algebra
$U(\mathfrak{g})$ of a finite-dimesional Lie algebra $\mathfrak{g}$ that is a
properly internalized version of a completed Hopf algebroid previously studied
as a Lie algebra type noncommutative phase space.
