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In this paper, we show that two constructions form stacks: Firstly, as onevaries the $\infty$-topos, $\mathcal{X}$, Lurie's homotopy theory of highercategories internal to $\mathcal{X}$ varies in such a way as to form a stackover the $\infty$-category of all $\infty$-topoi.  Secondly, we show that Haugseng's construction of the higher category ofiterated spans in a given $\infty$-topos (equipped with local systems) can beused 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 thediagram. For example, Riehl and Verity have shown that possessing (co)limits ofa given shape is hereditary. Extending their result somewhat, we show thatpossessing Kan extensions of a given type is heriditary, and more generallythat the adjointability of a functor is heriditary.

The stack of higher internal categories and stacks of iterated spans