I am currently following Scholze's lecture notes (https://www.math.uni-bonn.de/people/scholze/Condensed.pdf) on condensed mathematics. We would like to define condensed sets as sheaves on the site $\text{ProFin}$ of profinite spaces (with finite, jointly surjective families as covers). This poses set theoretic problems, as this site is essentially large, so we can not guarantee that $\text{Hom}$s in the sheaf category are sets (indeed such a morphism is a tuple of compatible maps, indexed by objects of the site), i. e. whether such a category of sheaves is locally small. So instead one tries to approximate this construction from below. We define the categories $\text{CondSet}_\kappa$ by consirdering sheaves on the smaller (now essentially small) site $\text{ProFin}_\kappa$ of $\kappa$-small profinite spaces, where $\kappa$ is an uncountable strong limit cardinal. This is indeed locally small. To identify objects in these categories for different $\kappa$, we consider a suitable transition/gluing maps $(-)^{\kappa\rightsquigarrow \kappa'}$ that turn out to be fully faithful (proposition 2.9). We can now try to glue these categories along transitions, and indeed in Definition 2.11, Scholze introduces condensed sets as the colimit $$\varinjlim_{\kappa}\text{CondSet}_\kappa$$ where the (large) colimit ranges over all uncountable strong limit cardinals $\kappa$, with the implied transition maps.
I am however cautiously sceptic about this definition. If we have three such cardinals $\kappa<\kappa'<\kappa''$ then to my understanding we only have $$(-)^{\kappa'\rightsquigarrow \kappa''}\circ (-)^{\kappa\rightsquigarrow\kappa'} \cong (-)^{\kappa\rightsquigarrow \kappa''}$$ not actual equality (even if we were to consider condensed sets as sheaves on extremally disconnected spaces and were allowed to drop the sheafification in the definition of $(-)^{\kappa\rightsquigarrow\kappa'}$). Hence the assignment $$\kappa\mapsto \text{CondSet}_\kappa,\;\;(\kappa<\kappa')\mapsto (-)^{\kappa\rightsquigarrow\kappa'}$$ actually fails to be a functor (the composition of morphisms $\kappa\to\kappa'$ and $\kappa'\to \kappa''$ under the assignment is not preserved). So it is not very clear to me, what is meant by a colimit of such an assignment.
I am (very) dimly aware of pseudo-/weak 2-functors that seem to capture this property of "being a functor up to natural equivalence" and allow a more general notion of (homotopy) colimit, I however have not worked through the details.
Immediately after this definition it is remarked (using that ($\kappa$-small) profinite spaces and extremally disconnected spaces give rise to the same sheaves) that one can think of condensed sets alternatively as being a full subcategory of sheaves $T$ of the naive $\text{Sh}(\text{ExDisc})$, which arise from some $\text{CondSet}_\kappa\simeq \text{Sh}(\text{ExDisc}_\kappa)$. More precisely such a naive sheaf $T$ is a condensed set, if it is isomorphic to the left Kan extension (which is essentially the inverse image of the map of sites $i\colon\text{ExDisc}\to\text{ExDisc}_\kappa$ corresponding to the inclusion) of its restriction to some $\text{ExDisc}_\kappa$ (i. e. the counit map $i^{-1}i_\ast T\to T$ is an isomorphism). Now denoting these left Kan extension functors as $(-)^{\kappa\rightsquigarrow\infty}:=i^{-1}$ (which share analogous properties to the functors $(-)^{\kappa\rightsquigarrow\kappa'}$, in particular are fully faithful), we find that condensed sets are the (full) union of the essential images of the functors $(-)^{\kappa\rightsquigarrow\infty}$ from $\kappa$-condensed sets to naive condensed sets. In particular any $\text{Hom}(F^{\kappa\rightsquigarrow\infty},G^{\kappa'\rightsquigarrow\infty})$ in this full (and a priori not necessarily locally small) category can be calculated instead in $\max(\kappa,\kappa')$-condensed sets by fully faithfulness, so a posteriori we have local smallness. This make me believe that one might as well just take this as the definition of condensed sets and never touch the colimit definition.
If one were to do this, I am still a bit unsure about set theoretic issues here. We initially introduced these cardinal bounds, so that we did not have to talk about the naive categories of sheaves at all. Now in the end it seems we still have to work with them somewhat, and only afterwards guarantee that for those objects we consider the $\text{Hom}$s are actually sets. This seems tricky and (at least to some degree) a bit dangerous to me; I am however not really well versed in the intricacies of set theory, especially when applied to category theory.
To my knowledge, what Clausen and Scholze are trying to archieve with this construction is to stay in the world of ZF+C. Otherwise one might instead try to work with Grothendieck universes to remedy these set-theoretic issues, at the cost of assuming a stronger system, like assuming the existence of inaccesible cardinals. Would this construction be "allowed" in ZF+C?
