Background: In functorial algebraic geometry one would like to consider the category of all functors $\mathsf{CRing} \to \mathsf{Set}$ and define/characterize the category of schemes as a full subcategory. However, there are serious set-theoretic difficulties: this is not a category, it is too large. One possible solution is suggested in Demazure-Gabriel's book. They use three nested universes. I wonder if we may stay in one universe. In general, one constructs the cocompletion $\widehat{\mathcal{C}}$ of a category $\mathcal{C}$ (not assumed to be small) as the category of functors $F : \mathcal{C}^{\mathrm{op}} \to \mathsf{Set}$ which are cofinally small, i.e. the category of elements $\int F$ is cofinally small. Then $\widehat{\mathcal{C}}$ is a category. We may apply this to $\mathcal{C}^{\mathrm{op}}=\mathsf{CRing}$ and develop functorial algebraic geometry inside $\widehat{\mathcal{C}}$. But the question is if the usual category of (geometric) schemes embeds into it. This leads to the following questions:
Question. Let $X$ be a scheme. Is there a set of commutative rings $\{A_i\}$ with the property that for every commutative ring $A$, every morphism $\mathrm{Spec}(A) \to X$ factors through some $\mathrm{Spec}(A_i)$?
The answer is yes if $X$ is quasi-compact quasi-separated. In fact, in that case $X(-) : \mathsf{CRing} \to \mathsf{Set}$ commutes with filtered colimits so that we may take a skeleton of all finitely generated commutative rings (which is a set). If $X$ is arbitrary, I guess that $X(-)$ commutes with $\lambda$-directed colimits for some cardinal number $\lambda$, which would answer the question. But I'm not sure about that.
Question. In case the answer to the first question is "Yes": Can we choose that set in such a way that for every $\mathrm{Spec}(A) \to X$ the category of morphisms $\mathrm{Spec}(A) \to \mathrm{Spec}(A_i)$ over $X$ is connected?
