(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
By categorification of the notion of geometric morphism, an $(\infty,1)$-geometric morphism is a pair of adjoint (∞,1)-functors between (∞,1)-toposes where the leftadjoint is left-exact.
For $\mathbf{H}$ and $\mathbf{K}$ two (∞,1)-toposes, a $(\infty,1)$-geometric morphism $f : \mathbf{H} \to \mathbf{K}$ is
an (∞,1)-functor $f_* : \mathbf{H} \to \mathbf{K}$
(called the direct image of the geometric morphism)
which has a left adjoint (∞,1)-functor $\mathbf{H} \leftarrow \mathbf{K} : f^*$;
(called the inverse image of the geometric morphism)
such that $f^*$ is a left exact (∞,1)-functor.
The (non-full) sub-(∞,1)-category of (∞,1)Cat on (∞,1)-toposes and $(\infty,1)$-geometric morphisms between them is (∞,1)Toposes.
For the moment see the discussion and the further links at geometric morphism
$(\infty,1)$-geometric morphism
section 6.3.1 in