This question prompted a reformulation:
What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit?
I believe this to be a serious question because it actually is oftentimes a good idea casually to identify isomorphism classes. To bring up an intermediate-level example I've alluded to often, consider the classification of topological surfaces. When I explain it to students, I do somewhat consciously write equalities as I manipulate one shape into another homeomorphic one. I even do it rather quickly to encourage intuitive associations that are likely to be useful. In any case, for arguments of that sort, it would be really tedious, and probably pointless, to write down isomorphisms with any precision.
Meanwhile, at other times, I've also joined in the chorus of criticism that greets the conflation of equality and isomorphism.
The problem is it's quite challenging to come up with really striking examples where this care is rewarded. Let me start off with a somewhat specialized class of examples. These come from descent theory. The setting is a map $$X\rightarrow Y,$$ which is usually submersive, in some sense suitable to the situation. You would like criteria for an object $V$ lying over $X$, say a fiber bundle, to arise as a pull-back of an object on $Y$. There is a range of formalism to deal with this problem, but I'll just mention two cases. One is when $Y=X/G$, the orbit space of a group action on $X$. For $V$ to be pulled-back from $Y$, we should have $g^*(V)\simeq V$ for each $g\in G$. But that's not enough. What is actually required is that there be a collection of isomorphisms $$f_g: g^*(V)\simeq V$$ that are compatible with the group structure. This means something like $$f_{gh}=f_g\circ f_h,$$ except you have to twist in an obvious way to take into account the correct domain. So you see, I have at least to introduce notation for the isomorphisms involved to formulate the right condition. In practice, when you want to construct something on $Y$ starting from something on $X$, you have to specify the $f_g$ rather precisely.
Another elementary case is when $X$ is an open covering $(U_i)$ of $Y$. Then an object on $Y$ is typically equivalent to a collection $V_i$ of objects, one on each $U_i$, but with additional data. Here as well, $V_i$ and $V_j$ obviously have to agree on the intersections. But that's again not enough. Rather there should be a collection of isomorphisms $$\phi_{ji}: V_i|U_i\cap U_j\simeq V_j|U_i\cap U_j$$
that are compatible on the triple overlaps:
$$\phi_{kj}\circ \phi_{ji}=\phi_{ki}.$$ Incidentally, for something like a vector bundle, since any two of the same rank are locally 'the same,' it's clear that keeping track of isomorphisms will be the key to the transition from collections of local objects to a global object. The formalism is concretely applied in situations where you can define some objects only locally, but would like to glue them together to get a global object. For a really definite example that comes immediately to mind, there is the determinant of cohomology for vector bundles on a family of varieties over a parameter space $Y$. Because a choice of resolution is involved in defining this determinant, which might exist only locally on $Y$, Knudsen and Mumford struggled quite a bit to show that the local constructions glue together. Then Grothendieck suggested the remedy of defining the determinant provisionally as a signed line bundle, which then allowed them to nail down the correct $\phi_{ji}$. These days, this determinant is a very widely useful tool, for example, in generating line bundles on moduli spaces.
I apologize if this last paragraph is a bit too convoluted for non-specialists. Part of my reason for writing it down is to illustrate that my main examples for bolstering the 'keep track of isomorphisms' paradigm are a bit too advanced for most undergraduates.
So, to conclude, I'd be quite happy to hear of better examples. As already suggested above, it would be nice to have them be accessible but substantively illuminating. If you would like to discuss, say, different bases for vector spaces, it would be good if the language of isomorphism etc. clarifies matters in a really obvious way, as opposed to a sets-and-elements exposition.
Added: Oh, if you have advanced examples, I would certainly like to hear about them as well.
Added: I see now there are three levels at least to distinguish:
Regarding objects as equal vs. regarding them as isomorphic vs. paying attention to specific isomorphisms.
I somehow conflated the two transitions in the course of asking the question. Of course I'm happy to see good examples illustrating the nature of either, but I'm especially interested in the second refinement.
Added yet again:: I'm grateful to everyone for contributing nice examples, and to Urs Schreiber who put in some effort to instruct me over at the n-category cafe. As I mentioned to Urs there, it would be especially nice to see examples of the following sort.
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One usually thinks $X=Y$;
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A careful analysis encourages the view $X\simeq Y$;
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This perspective leads to genuinely new insight and benefit.
Even better would be if some specific knowledge of the isomorphism in 2. is important. Of course, more than two objects might be involved. I was initially hoping for some input from combinatorics, with the emphasis on 'bijective proofs' and all that. Anything?
Added, 14 May:
OK, I hope this will be the last addition. Because this question flowed over to the n-category cafe, I ended up having a small discussion there as well. I thought I'd copy here my last response, in case anyone else is interested.
n-cafe post:
I suppose it's obvious by now that I'm using a specific request to drive home the need for 'small but striking examples' in favor of category theory.
Last fall, Eugenia Cheng told me of a visit to some university to give a colloquium talk. The host greeted her with the observation that he doesn't regard category theory as a field of research. OK, he was probably a bit extreme, but milder versions of that view are quite common. Now, one possible response is to regard all such people as unreasonable and talk just to friends (who of course are the reasonable people!). This is not entirely bad, because that might be a way to buy time and gain enough stability to eventually prove the earth-shattering result that will show everyone! Another way is to take up the skepticism as a constructive everyday challenge. This I suppose is what everyone here is doing at some level, anyways.
Other than the derived loop space, which is not exactly small, Urs' examples are all of the simple subtle sort that can, over time, contribute to a really important change in scientific outlook and maybe even the infrastructure of a truly glorious theory. For example, I agree wholeheartedly about the horrors of the old tensor formalism. But it's not unreasonable to ask for more striking accessible evidence of utility when it comes to the current state of category theory.
The importance of small insights and language that gradually accumulate into the edifice of a coherent and powerful theory is the usual interpretation of Grothendieck's 'rising sea' philosophy. However, the process is hardly ever smooth along the way, especially the question of acceptance by the community. I'm not a historian, but I've studied arithmetic geometry long enough to have some sense of the changing climate surrounding etale cohomology theory, for example, over the last several decades. The full proof of the Weil conjectures took a while to come about, as you know. Acceptance came slowly with many bits and pieces sporadically giving people the sense that all those subtleties and abstractions are really worthwhile. Fortunately, the rationality of the zeta function was proved early on. However, there was a pretty concrete earlier proof of that as well using $p$-adic analysis, so I doubt it would have been the big theorem that convinced everyone. One real breakthrough came in the late sixties when Deligne used etale cohomology to show that Ramanujan's conjecture on his tau function could be reduced to the Weil conjectures. There was no way to do this without etale cohomology and the conjecture in question concerned something very precise, the growth rate of natural arithmetic functions. This could even be checked numerically, so impressed people in the same way that experimental verification of a theoretical prediction does in physics. Clearly something deep was going on. Of course there were many other indications. The construction of entirely new representations of the Galois group of $\mathbb{Q}$ with very rich properties, the unification of Galois cohomology and topological cohomology, a clean interpretation of arithmetic duality theorems that gave a re-interpretation of class field theory, and so on.
For myself, being a fan of you folks here, I believe this kind of process is going on in category theory. But I don't think you have to be too unreasonable to doubt it. In a similar vein, I don't agree with Andrew Wiles' view that physics will be irrelevant for number theory, but also think his pessimism is perfectly sensible.
I think I'm trying to make the obvious point that the presence of pessimists can be very helpful to the development of a theory, in so far as the optimists interact with them in constructive ways. I haven't been coming to this site much lately, because the bit of internet time I have tends to be absorbed by Math Overflow. But I did catch David's recent post on Frank Quinn's article, which ended up as a catalyst for my MO question.
At the Boston conference following the proof of Fermat's last theorem, I've been told Hendrik Lenstra said something like this: 'When I was young, I knew I wanted to solve Diophantine equations. I also knew I didn't want to represent functors. Now I have to represent functors to solve Diophantine equations!' So should we conclude that he was foolish to avoid representable functors for so long? I wouldn't.
This response to the MO question brings up the importance of knowing the specific isomorphism between some Hilbert spaces given by the Fourier transform. This is an excellent example, especially when we consider how it relates to the different realizations of the representations of the Heisenberg group and the attendant global issues, say as you vary over a family of polarizations. But I couldn't resist recalling Irving Segal's insistence that 'There's only one Hilbert space!' Obviously, he knew, among many other things, the different realizations of the Stone-Von-Neumann representation as well as anyone, so you can take your own guess as to the reasoning behind that proclamation. He certainly may have lost something through that kind of philosophical intransigence. But I suspect that he, and many around him, gained something as well.
Best Answer
Suppose you have two categories C and D, and functors $F:C\to D$ and $G:D\to C$ such that for all $x\in C$, $G(F(x))$ is isomorphic to $x$, and for all $y\in D$, $F(G(y))$ is isomorphic to $y$. If you didn't know how important it was to distinguish between "isomorphic" and "an isomorphism," you might think that then C and D are essentially the same category. But of course in order for C and D to be equivalent, one additionally needs the isomorphisms in question to be natural.
A nice example of a pair of functor with the above properties, but which are not an equivalence of categories, are the functors relating vector spaces and affine spaces. Here $F:\mathrm{Vect}\to \mathrm{Aff}$ regards a vector space as an affine space by forgetting its origin, and $G:\mathrm{Aff}\to \mathrm{Vect}$ constructs the "vector space of displacements" in an affine space. The composite $G F$ is naturally isomorphic to the identity, but $F G$ is only "unnaturally" isomorphic to the identity, and the categories are definitely not equivalent. A simpler version of this example relates groups with heaps.