An interesting definition of a structure-preserving map inspired by natural transformation The book "Conceptual mathematics..." (Lawvere and Schanuel, 1997) introduces the category of "sets with endomaps" (that resemble a set theoretic representation of monoids). Then the book proceeds to give the definition of a "structure-preserving maps" between those structures that conceptually resemble functors. However, the definition of this map is actually quite different from the traditional functor definition and is actually closer to the definition of natural transformation (see the screenshot.)
Later the book also reviews the category of (set-theoretic representations of) graphs and the definition of maps between them is similar.
I was wondering what is the formal definition of such structure-preserving maps would be and if they are defined anywhere else.

 A: There are many versions in many abstraction levels of 'structure preserving maps'.
A 'set with endomap' in universal algebraic language is called a 'unary algebra': it is a set $X$ with a single unary operation $\alpha:X\to X$.
Further universal algebraic structures might have multiple finitary operations (an $n$-ary operation is simply a map $X^n\to X$).
Further, there are first order structures. Those themselves have multiple notions of 'structure preserving maps' because of the presence of relations.
A category theoretic approach is as follows:
Consider an endofunctor $T:\mathcal S\to\mathcal S$ on a category $\mathcal S$ (which is typically $\mathcal{Set}$).
Then a $T$-algebra is a pair $(x,\alpha)$ where $x\in Ob\,\mathcal S$ and $\alpha:T(x)\to x$.
A structure preserving map from $(x,\alpha)$ to $(y,\beta)$ is an arrow $f:x\to y$ in $\mathcal S$ which satisfies $\beta\circ T(f)=f\circ\alpha$.
Often, $T$ is assumed to be a monad and then the algebra structure also needs to be compatible with the monad structure (see section 'algebras over a monad').
