I Prove Something is a Category. Now What? So I am just getting more into category theory, and particularly curious about the applied branches.  So this question might come off as terribly naive, since I have not had much exposure to category theory in general.
So let's say I have some phenomenon, system, etc. that comes from some applied setting (right now I don't care which setting it is, could be physics, economics, organizational design...whatever, just something in the "real world").  And let's say that I see that this object, or the set of these objects, form a category.
But what does that actually get me?  Is there a next step that I can take, where noticing that something is a category (or a specific type of category) that is not just a restatement or a neat fact to know?
For example, I have been perusing John Baez's website.  One thing that comes up is that things such as electrical circuits form a strict symmetric monoidal category.  Does this fact tell me anything I can do/analyze with circuits that I couldn't do before (with, say, knowledge of physics or engineering usually taught at the undergrad level), because all that I have gotten in my (very limited) look at the subject is "hey these can be looked at as a category, ain't that pretty neat."
 A: The comment about "all the theorems of category theory" is quite misleading, for the following reason. A very important thing about category theory which no one will tell you is that there are, roughly speaking, two quite different kinds of categories, and the "theorems of category theory" as such mostly concern the second kind. Informally, they are

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*"little" categories: examples include the category $BG$ with one object and automorphism group $G$, or the free category on a directed graph. "Little" categories tend to be small (set-sized) or even finite, and rarely or never have interesting limits or colimits. Their role is usually to serve as shapes for diagrams; in other words, the point of "little" categories is usually that we consider functors out of them.

*"Big" categories: examples include the category $\text{Set}$ of sets, or other categories of mathematical objects such as the categories of groups, rings, or topological spaces. "Big" categories tend to be large (not set-sized), and typically have interesting limits and colimits. These are the categories to which the interesting theorems of category theory as such, e.g. the adjoint functor theorem, actually apply, and they can be both the source and the target of interesting functors.

There are essentially no interesting theorems about "little" categories (without significant further hypotheses, e.g. in the theory of fusion categories), at least not that I know of, and the categories that show up in Baez's network stuff are "little" categories, so just knowing that electrical networks form a monoidal category by itself doesn't tell you much. The point of knowing this is that it suggests category-flavored questions: for example, if electrical networks form a (monoidal) category, what can we say about (monoidal) functors out of this category? Such functors would allow us to analyze electrical networks by decomposing them into parts and then composing and tensoring the parts together; this is a categorical way to reason about "compositionality."
I'm not familiar enough with this specific network stuff to give any examples specific to networks, unfortunately. But Baez is inspired here by similar ideas from quantum field theory and the circle of ideas around Penrose graphical notation and topological quantum field theory and so forth, where monoidal categories feature prominently; you can check out e.g. his Physics, Topology, Logic, and Computation: A Rosetta Stone for some more discussion of this circle of ideas.
