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From the comments to this question I have learned, that many (most?) mathematicians are not very interested in the relationship between an object $X$ and its "correspondent" $F(X)$ for an arbitrary (equivalence) functor $F$, at least not interested enough to name it. Opinions range from "it's not usual" over "people don't miss it" to "one cannot really compare $X$ with $F(X)$". From that I conclude that – especially – one would not (always) call $X$ and $F(X)$ essentially the same, not even for an equivalence functor $F$.

The non-technical term essentially the same (e.t.s.) has at least two (pedagogical) usages in the context of category theory: (i) two isomorphic objects are e.t.s. and (ii) two equivalent categories are e.t.s.

Outside of category theory one might for example say that corresponding vertices of a graph isomorphism are e.t.s. because they are indistinguishable by their "network of relationships". Or conjugate members of a group. Accordingly, one might say that corresponding objects of an isomorphim between categories are e.t.s.

I believe - but I may be mistaken - that at least some mathematicians would say that at least some non-isomorphic functors give rise to "essentially the same" objects, e.g. in the context of representations: a Boolean algebra $B$ is essentially the same as its Stone space $S(B)$. (Or is this too far-fetched, and you would never say this?)

Another example: Consider the functor $F: \mathsf{Graph} \rightarrow \mathsf{Graph}$ from the standard category of simple graphs with graph homomorphisms to itself which adds an extra node on each edge. You can interpret this functor like this: the new nodes represent the original edges, the new edges represent incidence of original nodes and edges. At least me, I'd definitely say that a graph $\Gamma$ and its image $F(\Gamma)$ are essentially the same.

Under which circumstances (conditions on the categories and functors involved) is one willing to call $X$ and $F(X)$ essentially the same.

Please note that - despite the title of this question - I am not asking for a general definition for two arbitrary mathematical objects to be essentially the same which would subsume questions like "When are two algorithms the same?" or "When are two proofs the same?". I only ask for conditions under which one is willing to speak of "essential sameness" in the context of category theory.

PS: I must not conceal the origin of my question's title. It's - undeniably - Mazur's When is one thing equal to some other thing? (even though Mazur doesn't use the words "essentially the same").

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I highlighted the sentence: "I only ask for conditions under which..." to help clarify your question. If you'd prefer, you (or I) can "roll back" to your most recent edit. On a different note, see equivalence of categories. –  amWhy Nov 13 '12 at 13:57
@amWhy: If you find it more readable and concise this way, I am happy with it. –  Hans Stricker Nov 13 '12 at 14:01
Alternatively, I simply italicized the sentence, to help hone in on what you're asking. That way, your formatting is better preserved. +1, by the way. –  amWhy Nov 13 '12 at 14:05
Your questions seem to converge fast to philosophy:) –  Berci Nov 13 '12 at 15:03
I do my best to keep them in touch with mathematics:) –  Hans Stricker Nov 13 '12 at 15:50

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