Let $C$ be a locally small category. Then we have the Yoneda embedding $Y : C \to \widehat{C} := [C^{op},Set]$. Recall that $C$ is called total when $Y$ has a left adjoint.

My first question is: Why aren't there any set-theoretic obstructions for such a left adjoint to exist? The category $\widehat{C}$ is not locally small, lives in a bigger universe than $C$. In some papers I've read that, for instance, algebraic categories are total. Ok, but how does the left adjoint of $Y$ looks like *explicitly* for $C=\mathsf{Ring}$ for instance? I'm not yet convinced that such a functor can exist ...

The second question is: What do you think of the following variant of totality (and is it already known and studied in the literature): Define $\widehat{C}$ to be the category of presheaves on $C$ which are small colimits of representable functors. Equivalently, these presheaves are cofinally small. Then $\widehat{C}$ lives in the same universe of $C$, and in fact the Yoneda embedding enjoys the universal property of the universal cococompletion of $C$ (see here).

In this setting, I can show that the category of affine schemes $\mathsf{Aff}$ is total (i.e. $\mathsf{Ring}$ is cototal), and in fact the corresponding adjunction is quite well-known: To every affine scheme $X$ we associate its functor of points $\hom(-,X)$. To every cofinally small presheaf on affine schemes $F$ we associate $\mathcal{O}(F) := \hom(F,\mathbb{A}^1)$ with component-wise ring operations (or rather we associate $\mathrm{Spec} \mathcal{O}(F)$). Note that we could also do this construction for every presheaf $F$ (and this is done in many texts, for example in J. S. Milne's script on algebraic groups), but then $\mathcal{O}(F)$ has no chance of being a small set and therefore an object of $\mathsf{Ring}$.