Top of p.iv, "almost no introductory material available on monoidal categories." Wait, it was all about categories, now it is suddenly monoidal ones -- what is monoidal, is that a subset of the book subject, or a more precise term? Badly need a definition when first introducing a new term.
(Actually, that just highlights the point that the term "category" has been assumed and not defined to this point -- despite the audience not being expected to know what it is.)
Lemon pie diagram: of _course_ I can grasp the basic idea, but in what sense does the diagram illustrate category theory? Where are is the category (categories?), is it the pie, the lines, the nodes? It's probably blindingly obvious _after you understand the topic_ but at the opening of the book, the point needs to be explained even if the explanation seems redundant to the author.
Top paragraph on p.v flirts with a definition but doesn't actually define anything. Is a category "structures and coherence"? One "involves" (vague) "a collection of objects, a collection of morphisms relating objects, and a formula for combining any chain of morphisms into a morphism" and OK, here is a hint of familiar ground: this sounds a lot like category theory might be a generalization of classes, instances and methods, and functional composition, of software engineering. But the sentence about chains of chains ending in a happy "That's it!" doesn't really seem to follow. Onward.
Bottom of page 1, we finally get to some substance, although I really don't like the "Y is a meter" explanation because that whole paragraph kind of bodges the distinction between italic-capital-letter X and Y, which I am kinda sure you intend to represent classes or domains, with the word "object" which my prior training really wants to confine to singular instances of a domain. So what comes out of the map isn't a "meter Y" but "the reading displayed by the meter Y" but that doesn't work either.
Anyway, very shortly you drop in another new notation, "functions f from big-R to big-R" What does the big-R mean here? I'm casting about -- real-world-objects? set of Real numbers? Probably real numbers because the following features are about arithmetic comparisons and operations. But say it, dammit.
Order-preserving, ok, but -- why "metric-preserving"? How is absolute-difference a "metric"? But... this is an exercise and a challenging one, and I'm intrigued. Later.
I think that the best way forward in your case is simply googling terms that the writers took for granted and which you don't understand (a monoid, for example), because the book material is one of the best I've seen on category theory.
I recall that my personal experience of learning about CT was unnecessarily difficult due to being thrown into a bunch of definitions way to soon. (Which is understandable since there isn’t much else to it.) but I needed to un-learn my preconceptions about what information the theory encodes first, which was rally hard. Understanding that objects and morphisms are really _only_ about composability and _nothing else_ is challenging when your mind desperatly wants to see concrete thing like sets or numbers or what have you in those objects. Definition don’t help much until you can shed that urge. (To new learners I suggest to think of “objects” as the number of dots on domino tiles, or the poles of magnets, they describe where things may, or may not, compose, nothing more. category theory is just about naming patterns and rules of composition)
In particular even knowing the existence of monoidal categories is a distraction when getting to grips with how catgories as such generalize and abstracts monoids.
I had a similar experience. Luckily, due to reading many definitions of the basic things (such as monoids), I reached a more complete? intuition of the concept if you will. I think that given enough definitions, people will converge to the same intuition of abstract concepts.
> To new learners I suggest to think of “objects” as the number of dots on domino tiles, or the poles of magnets, they describe where things may, or may not, compose, nothing more
Wouldn't a simpler intuition of circles and arrows be easier?
Categories might be drawn using arrows and circles. But that’s just notation.