  In this paper, we show that two constructions form stacks: Firstly, as one
varies the $\infty$-topos, $\mathcal{X}$, Lurie's homotopy theory of higher
categories internal to $\mathcal{X}$ varies in such a way as to form a stack
over the $\infty$-category of all $\infty$-topoi.
  Secondly, we show that Haugseng's construction of the higher category of
iterated spans in a given $\infty$-topos (equipped with local systems) can be
used to define various stacks over that $\infty$-topos.
  As a prerequisite to these results, we discuss properties which limits of
$\infty$-categories inherit from the $\infty$-categories comprising the
diagram. For example, Riehl and Verity have shown that possessing (co)limits of
a given shape is hereditary. Extending their result somewhat, we show that
possessing Kan extensions of a given type is heriditary, and more generally
that the adjointability of a functor is heriditary.
