  We show that categories of modules over a ring in Homotopy Type Theory (HoTT)
satisfy the internal versions of the AB axioms from homological algebra. The
main subtlety lies in proving AB4, which is that coproducts indexed by
arbitrary sets are left-exact. To prove this, we replace a set with its strict
category of (ordered) finite sub-multisets. From showing that the latter is
filtered, we deduce left-exactness of the coproduct. More generally, we show
that exactness of filtered colimits (AB5) implies AB4 for any abelian category
in HoTT. Our approach is heavily inspired by Roswitha Harting's construction of
the internal coproduct of abelian groups in an elementary topos with a natural
numbers object [Har82].
  To state the AB axioms we define and study filtered (and sifted)
precategories in HoTT. A key result needed is that filtered colimits commute
with finite limits of sets. This is a familiar classical result, but has not
previously been checked in our setting.
  Finally, we interpret our most central results into an $\infty$-topos
$\mathscr{X}$. Given a ring $R$ in $\mathscr{X}$, we show that the internal
category of $R$-modules in $\mathscr{X}$ represents the presheaf which sends an
object $X \in \mathscr{X}$ to the category of $(X{\times}R)$-modules over $X$.
In general, our results yield a product-preserving left adjoint to base change
of modules over $X$. When $X$ is $0$-truncated, this left adjoint is the
internal coproduct. By an internalisation procedure, we deduce left-exactness
of the internal coproduct as an ordinary functor from its internal
left-exactness coming from HoTT.
