The category $\mathbf{Cond}$ of condensed sets is equivalent to the category of small sheaves over any of the following three large sites. (For small sheaves, see Mike Shulman's paper Exact completions and small sheaves in TAC.)

- The category $\mathbf{Comp}$ of compact Hausdorff spaces, equipped with the coherent topology (i.e. finite jointly epimorphic families are covering);
- The category $\mathbf{ProFin}$ of profinite sets, equipped with the coherent topology;
- The category $\mathbf{Proj}$ of projective compact Hausdorff spaces, equipped with the disjunctive topology (i.e. families of finite coproduct inclusions are covering).

It follows that for any Grothendieck topos (and more generally any infinitary-pretopos) $\mathscr{E}$, the category of cocontinuous left exact functors from $\mathbf{Cond}$ to $\mathscr{E}$ is equivalent to the following three categories:

- The category of coherent functors from $\mathbf{Comp}$ to $\mathscr{E}$ (i.e. functors that preserve finite limits, finite coproducts, and epimorphisms);
- The category of coherent functors from $\mathbf{ProFin}$ to $\mathscr{E}$;
- The category of functors from $\mathbf{Proj}$ to $\mathscr{E}$ that preserve finite coproducts and weak finite limits.

These are all instances of the universal property of the category of small sheaves on a (possibly large) site (for which, see Shulman's paper cited above). Namely, the category of condensed sets is the *infinitary-pretopos completion* of (i) the pretopos $\mathbf{Comp}$, (ii) the coherent category $\mathbf{ProFin}$, (iii) the weakly lextensive category $\mathbf{Proj}$.

Now, essentially because the category $\mathbf{Cond}$ of condensed sets is "too big" (e.g. it is not locally presentable), cocontinuous functors $\mathbf{Cond} \to \mathscr{E}$ need not have right adjoints. For that reason, in place of geometric morphisms (defined as adjoint pairs with left exact left adjoint) from a Grothendieck topos $\mathscr{E}$ to $\mathbf{Cond}$, one should instead consider cocontinuous left exact functors from $\mathbf{Cond}$ to $\mathscr{E}$. In particular, the "correct" notion of *point* of $\mathbf{Cond}$ is a cocontinuous left exact functor $\mathbf{Cond} \to \mathbf{Set}$.

Thus, by the above, points of $\mathbf{Cond}$ are equivalent to (i) coherent functors $\mathbf{Comp} \to \mathbf{Set}$, (ii) coherent functors $\mathbf{ProFin} \to \mathbf{Set}$, and (iii) functors $\mathbf{Proj} \to \mathbf{Set}$ that preserve finite coproducts and weak finite limits.

Finally, for $\kappa$ a strong limit cardinal, the category of $\kappa$-condensed sets is equivalent to be the category of sheaves over the sites $\mathbf{Comp}_{\kappa}$, $\mathbf{ProFin}_{\kappa}$, and $\mathbf{Proj}_{\kappa}$, defined to be the full subcategories of the above sites spanned by the spaces of cardinality $< \kappa$, with the induced topologies. As above, it follows that points of the topos of $\kappa$-condensed sets are equivalent to certain kinds of functors (the same kinds as above) from these full subcategories to $\mathbf{Set}$.