What was the number one thing about category theory that changed your perspective if you don’t mind me asking? You seem very enthusiastic and I’d like to share that enthusiasm one day.
And it can be useful to think of them more abstractly than they actually are, even if you're not going to use them that way (or even talk about them that way in your code). Just making some "ridiculously abstract" connections can help provide a missing piece of architecture, even if you keep the math story to yourself (which you likely should).
So it can (potentially?) be useful to model various things in terms of what they mean "at the most abstract level", and then find ways to capture chunks of that into various architectural abstractions that make sense without category theory (kind of like a Karnaugh map, in a sense).
Sorry if that seems like a lot of "purple prose" or self-important... it really isn't. I think it's the same thing that a lot of people mean when they say "I don't use Haskell but learning it made me a better programmer." I don't use abstract algebra, but learning it made me a better programmer (those statements might even be homomorphic, depending on the Haskell involved)
Whiteboard diagrams work because they’re a categorical diagram related to the semantics of the program I’m trying to write.
In the sense that CT and TT are “equivalent”, your whiteboard diagrams are your program.
This genre of abstraction is all over the place in category theory and it gets way more interesting, but seeing products defined like this and how incredibly unifying of an abstraction it is was the first place that I really saw what the categorical perspective brought to the table.
*Cartesian product within the category of sets and functions between them (amounting to multiplication of cardinals, if one just cares about the action on objects), least common multiple within the category of positive integers ordered by divisibility (a partial ordering being just a special kind of category), logical conjunction within the category of truth values (which can be thought of as sets with at most one element), and structs or record types in the category whose objects are the types of your favorite programming language and morphisms are the programs between them. (From Chinjut, last time I brought up categorical products on hn: https://news.ycombinator.com/item?id=8780786)
[0] https://rnhmjoj.github.io/category-theory-for-programmers/sr...
If you'd to see how this applies more directly to programming, work like this is a direct application of the same definitional strategy for abstraction: https://blog.sumtypeofway.com/posts/introduction-to-recursio...
I wrote up a bit on it here https://github.com/adamnemecek/adjoint