@ElectricPenguin Thanks. Interesting point. I do question your assertion that there is more skill at Mathematics for solving such problems, but next time I see a case where this community finds a definite integral question on-topic and is defeated by it, I'll think about it. But based on a long history, it seems user64494 says the same thing just about every time he sees a definite integral problem for which Mathematica gives an answer (and, tbh, user's formulation sounds a bit rhetorical to me, as if asking "shouldn't you know better?", thus prompting my comment).

This may be reopened after the bounty expires.

I'm voting to close this question as off-topic because there is still a bounty on the question at Mathematics.

@user64494 I agree with Yemon's response. Even worse, such comments serve to "shame" the OP for even asking here, when in fact the question is perfectly on-topic, for the reasons Yemon gave. Please reflect carefully on this.

More readable, huh. Well, I first learned it from Categories for the Working Mathematician; you could try there. It's probably in a bunch of places, but I am not intimately familiar with the more recent introductory textbooks. But (since I had a certain amount to do with the writing of the nLab article): which technical details lost you, or where did you first find yourself getting lost?

@BjørnKjos-Hanssen You'd be in the best position to judge which post best answers your question or helps you the most, so I defer to you.

This post, while useful in some ways (and not really convertible to a comment), has been flagged "not an answer". Maybe, as a compromise, make it CW?

(No, it's $f_\bullet$ preserves sups.) Do you know the adjoint functor theorem, which generalizes this formula? ncatlab.org/nlab/show/adjoint+functor+theorem

My answer has just been downvoted. I invite the downvoter to explain why.

@j.c. The link to the Kalman pdf file seems to be broken.

I am having trouble understanding this post. Let's start with Card($Y_x$). If you had said nothing more, then (in Set say) I would suppose you mean you start with a map $p: Y \to X$ with finite fibers, and form a classifying map $\chi_p: X \to \mathbb{N}$ so that $\chi_p(x) = $ Card($Y_x$). (The classifying bundle is the map $E= \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ that takes a pair $(m, n)$ to $m+n+1$, so that $E_k$ has $k$ elements.) If this is on the right track, then how is it connected with what you wrote?

Well, it seems what Tony means is that he doesn't know what Doubilet, Rota, and Stanley meant when they wrote that. Which, if we read your question, seems to be what you were asking.

I'm not aware of any such modification, and I kind of doubt it since you'd still need conditions on monads/comonads on $\mathrm{Top}$ that come from the string of adjunctions.