Just as some motivating background in case there is any extra light you can shed on the topic beyond the specifics of the question, the reason why I'm looking at this is I'd heard vaguely about Lawvere's contributions to modal logic, and some weird invocations of Hegelian philosophy and wanted to understand what that was all about, so I've been reading up on the related Lawvere papers. One of the things I'm trying to get is adjoint cylinders/adjoint modalities/unity of opposites. The toy example on nlab is intuitive enough, I can check that the natural transformations all checkout and what not. I start to fumble after that. Now onto the particular problem at hand!
In one of Lawvere's papers on adjoint cylinders, he includes as an example a formalization of ideas previously articulated in non-categorical terms by Hadamard and Marx. I'm having trouble understanding even the beginning of this example. As I understand it we start by considering three ring homomorphisms and two commutative rings arranged such that one retracts the other two. So for example this might look like: $f, g : R_1 \to R_2$, $h : R_2 \to R_1$ where $R_1$ and $R_2$ are commutative rings and $h$ retracts $f$ and $g$. He then moves to talking about the case of smooth algebras which is where I start to get fuzzy as we start to deal with concepts I'm not well versed with. Besides generally knowing that smooth manifolds/algebras are where we do calculus I don't know much else. So when he starts talking about a quantity $y$ in the smaller ring becoming a quantity $y_0$ and $y_1$ I have no idea what he is talking about. Also the statement that the morphisms $M \to M \times M$ indicate the two trivial ways of sending a function of one variable to a function of two variables by composing with respective projections doesn't make sense to me. I don't understand how being able to convert $R \to A$ into $R \times R \to A$ yields the homorphisms we want. This is all before getting to the later issues...
If $y\in R_1,$ then $y_0$ and $y_1$ are $f(y)$ and $g(y).$ For instance, if $R_1$ is a ring of functions on a space $M$ and $R_2$ the corresponding ring of functions on $M\times M,$ then $y$ is a function of a single variable running over $M$ while $y_0$ and $y_1$ are functions of two variables, in the trivial sense that $y_0(m_0,m_1)=y(m_0)$ and $y_1(m_0,m_1)=y(m_1).$ You've shifted notations twice during your question, so I'm not entirely clear whether you're asking about something different in the line about $R$ and $A.$