The problem is that you cannot choose a domain and codomain for such a putative adjunction consistently and simultaneously. The statement that we have is that the category of presheaves on $C$ is the free cocomplete category on $C$ when $C$ is small. However, the forgetful functor from cocomplete categories to categories does not land in small categories-every cocomplete category which is not a preorder is large.
So, you might want an improved adjunction between large categories and cocomplete categories. However, the category of presheaves on a large category is even larger than large! What this means depends on your foundations. If we work with universes so that a small category is $U_1$-small, then presheaves on a small category are small with respect to the next biggest universe $U_2$. Now presheaves on a $U_2$-small category only have the appropriate universal property if we can them to be presheaves of $U_2$-small sets, and such presheaves are not $U_2$-small.
So the presheaves-forgetful functor pair cannot form an adjunction, because the desired left adjoint moves us up a universe every time we apply it. Thus in particular we cannot get around this straightforwardly by using universes.
There are a couple of partial solutions to this problem. The simplest is to regard the formation of the presheaf category as a relative left adjoint to the forgetful functor from cocomplete categories to (possibly) large categories. In other words, it behaves like a left adjoint, but is only partially defined. This is a rephrasing of the theorem you quote-the formation of presheaves behaves like a left adjoint when its input is small.
A more technical approach is to ask the question: even if the formation of presheaves cannot be left adjoint to the forgetful functor from cocomplete categories to large categories, does this forgetful functor have any left adjoint at all? In fact it does; for instance, it satisfies a 2-categorical version of the general adjoint functor theorem. This left adjoint sends a category $K$ to the subcategory of presheaves on $K$ formed by the colimits of small diagrams of representable presheaves. However, this category of small presheaves is not nearly as well behaved as the presheaf category. It needn't even be a topos in general.
So to summarize, this is a real issue which cannot be eradicated by any level of generous assumptions on the foundations. It's the go-to example of why size issues cannot be completely ignored in category theory.
If your goal is to read Demazure and Gabriel's book, then (as I explained in the comments) your question is based on false premises and the solution is to read the definitions carefully.
But let me address your question as written, since it will illuminate why Demazure and Gabriel use the definitions they do.
First, as you observe, the category of elements of an arbitrary functor $F : \textbf{CRing} \to \textbf{Set}$ is not always small.
Actually, it is almost never small, because $\textbf{CRing}$ itself is not small: as soon as $F (A)$ is non-empty for all rings $A$, then the category of elements of $F$ will be at least as big as $\textbf{CRing}$.
This is not actually fatal for the problem at hand (though it does introduce many complications).
It sometimes happen that the functor $F$ you are interested in is a colimit of a small diagram of representable functors, i.e. there is a small diagram $A : \mathcal{I}^\textrm{op} \to \textbf{CRing}$ such that $F (B) \cong \varinjlim_\mathcal{I} \textbf{CRing} (A, B)$ naturally in $B$.
In that case, you can compute the colimit $\left| F \right|$ you seek as $\varinjlim_\mathcal{I} \operatorname{Spec} A$.
The biggest complication is that $\left| F \right|$ is not well defined for arbitrary $F$, so you only get a partially defined functor $\left| - \right|$.
If you try to restrict to a full subcategory of functors $\textbf{CRing} \to \textbf{Set}$ on which $\left| - \right|$ is well defined everywhere, you then have the complication that the putative right adjoint may not have image contained in that subcategory.
I am not aware of any good way to resolve this dilemma; I think you have no choice but to settle for a partially defined adjoint.
Now for some good news: there is a clean necessary and sufficient condition for a functor $F : \textbf{CRing} \to \textbf{Set}$ to be a colimit of a small diagram of representable functors.
Definition.
Let $\kappa$ be an infinite regular cardinal.
A $\kappa$-accessible functor is a functor that preserves $\kappa$-filtered colimits.
Proposition.
Let $F : \textbf{CRing} \to \textbf{Set}$ be a functor.
The following are equivalent:
- $F$ is $\kappa$-accessible.
- $F$ is the left Kan extension of a functor $\textbf{CRing}_\kappa \to \textbf{Set}$ along the inclusion $\textbf{CRing}_\kappa \hookrightarrow \textbf{CRing}$, where $\textbf{CRing}_\kappa$ is the full subcategory of $\kappa$-presentable rings (i.e. rings presentable by $< \kappa$ generators and $< \kappa$ relations).
- There is a small diagram $A : \mathcal{I}^\textrm{op} \to \textbf{CRing}$ such that $F \cong \varinjlim_\mathcal{I} \textbf{CRing} (A, -)$ and, for each $i$ in $\mathcal{I}$, $A (i)$ is a $\kappa$-presentable ring.
The functor $R \mapsto R^{\oplus \mathbb{N}}$ you mention is easily seen to preserve filtered colimits (i.e. be an $\aleph_0$-accessible functor).
It is just as easy to see that it is the colimit of a small (indeed, countable!) diagram of representable functors, namely,
$$\textbf{CRing} (\mathbb{Z}, -) \longrightarrow \textbf{CRing} (\mathbb{Z} [x_1], -) \longrightarrow \textbf{CRing} (\mathbb{Z} [x_1, x_2], -) \longrightarrow \cdots$$
where the maps are the ones induced by the homomorphisms $\mathbb{Z} [x_1, \ldots, x_n, x_{n+1}] \to \mathbb{Z} [x_1, \ldots, x_n]$ that send $x_i$ to $x_i$ for $1 \le i \le n$ and $x_{n+1}$ to $0$.
Thus, the geometric realisation of $R \mapsto R^{\oplus \mathbb{N}}$ is the colimit $\varinjlim_n \mathbb{A}^n$.
I suppose I owe you an example of a functor $\textbf{CRing} \to \textbf{Set}$ that is not accessible.
Choose an ordinal-indexed sequence of fields, $K_\alpha$, such that $K_\alpha$ is strictly smaller in cardinality than $K_\beta$ whenever $\alpha < \beta$.
Let $F (R) = \coprod_{\alpha} \textbf{CRing} (K_\alpha, R)$ for non-zero rings $R$ and let $F (\{ 0 \}) = 1$.
Since any ring homomorphism $K_\alpha \to R$ is injective when $R$ is non-zero, $\textbf{CRing} (K_\alpha, R)$ is empty for sufficiently large $\alpha$, so $F (R)$ is indeed a set.
On the other hand, it is clear that $F$ cannot be the left Kan extension of any functor $\textbf{CRing}_\kappa \to \textbf{Set}$: if it were, it would be impossible to distinguish between this $F$ and the one where we cut off the disjoint union at some ordinal $\beta$ such that $K_\beta$ is not $\kappa$-presentable.
Best Answer
If $H \subseteq G$ is a subgroup and $X$ is a $G$-set (meaning: left $G$-set), then $G$-maps $G/H \to X$ correspond naturally to elements $x \in X$ with $H \subseteq G_x$. (You can easily write down a bijection.) This describes the hom-functor $\hom(G/H,-)$ on the category of $G$-sets. Notice that it is (isomorphic) to a subfunctor of the forgetful functor $U$, since what we get above is a subset of (the underlying set of) $X$. This means that when we take colimits of such functors, we are actually just taking a union inside the partial order of subfunctors. So, the canonical map
$\mathrm{colim}_{H \subseteq G \text{ finite index}} \hom(G/H,X) \to U(X)$
is injective, and the image consists of those elements $x \in X$ such that there is a finite index subgroup $H$ with $H \subseteq G_x$. Equivalently, $G_x$ must be of finite index.
Let $X$ be a finite $G$-set. Then for every $x \in X$ the stabilizer $G_x$ is of finite index (the index is the size of the orbit). Hence,
$\mathrm{colim}_{H \subseteq G \text{ finite index}} \hom(G/H,X) \to U(X)$
is bijective. We get an isomorphism of functors
$\mathrm{colim}_{H \subseteq G \text{ finite index}} \hom(G/H,-) \xrightarrow{\cong} U,$
where $U : \mathbf{FinSet}_G \to \mathbf{Set}$ is the forgetful functor.
We want the same with normal subgroups. Well, if $x \in X$, there is always a finite index normal subgroup $N \subseteq G_x$, namely $N = \bigcap_{g \in G} g G_x g^{-1} = \bigcap_{y \in Gx} G_{y}$.
I just write $N$ in the following to indicate a normal subgroup. Therefore, we get a bijection: $$\hom(U,U) \cong \hom\bigl(\mathrm{colim}_{N \subseteq G \text{ finite index}} \hom(G/N,-),U\bigr)\\ \cong \lim_{N \subseteq G \text{ finite index}} \hom\bigl(\hom(G/N,-),U\bigr)$$ The next step is to use the Yoneda Lemma. It implies that the natural map $$\hom\bigl(\hom(G/N,-),U\bigr) \to U(G/N)$$ is an isomorphism. Thus, we get $$\hom(U,U) \cong \lim_{N \subseteq G \text{ finite index}} U(G/N).$$ The next step is to upgrade this to an isomorphism of monoids $$\mathrm{End}(U) \cong \lim_{H \subseteq G \text{ finite index}} G/N.$$ This is not completely formal, the above argument is not symmetric! To see this, one just has to go through the proof and find an explicit formula for the bijection. It maps a natural transformation $\eta : U \to U$ to the compatible family of elements $\eta_{G/N}([1])$, where $[1] \in G/N$ is the canonical element. Conversely, if a compatible family $([g^N] \in G/N)_{N}$ is given, $\eta_X : U(X) \to U(X)$ is defined as follows: If $x \in X$, then $\eta_X(x) = [g^{G_x} \cdot x]$. For $X = G/N$, which also has a right $G/N$-action, this shows that $\eta_X$ is actually compatible with the right action*.
Using this, if $\eta,\mu : U \to U$, then $\eta \circ \mu$ is mapped to the family of elements $$\eta_{G/N}(\mu_{G/N}([1])) = \eta_{G/N}([1] \cdot \mu_{G/N}([1])) = \eta_{G/N}([1]) \cdot \mu_{G/N}([1]).$$ And clearly $\mathrm{id} : U \to U$ is mapped to the family $([1])$, which is $1$. This proves the isomorphism of monoids.
A projective limit of monoids, which happen to be groups, is again a monoid which happens to be a group. (This is basically because the compatibility condition transfers to inverse elements.) So the projective limit above is a group, and hence the monoid $\mathrm{End}(U)$ is a group. So it must coincide with its group of units, which is $\mathrm{Aut}(U)$. Also, an isomorphism of monoids between monoids which happen to be groups is actually an isomorphism of groups (it is well-known that we get preservation of inverses for free). So we indeed get an isomorphism of groups $$\mathrm{Aut}(U) \cong \lim_{N \subseteq G \text{ finite index}} G/N.$$ Instead of $\mathbf{FinSet}_G$ we can also the the category of $G$-sets whose stabilizers are of finite index (equivalently, all orbits must be finite).
*Remember that, even though $\eta$ is indexed by left $G$-sets, each map only operates between their underlying sets. The map has no reason to be a left $G$-map. It is a pure "coincidence" (as always, more general categorical arguments reveal that it is no coincidence) that sometimes these maps happen to be right $G$-maps, though. The natural transformations that consist of left $G$-maps are quite boring, this is $\mathrm{End}(\mathrm{id}_{\mathbf{Set}_G}) \cong Z(G)$, the center of $G$.