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Two categories C and D are equivalent if there exist functors F: C → D and G: D → C with natural isomorphisms GF ≅ Id_C and FG ≅ Id_D. Equivalence is weaker than isomorphism (which requires GF = Id_C exactly) but is the correct notion of 'sameness' for categories, since categorical properties are invariant under equivalence. A functor F: C → D is an equivalence if and only if it is full, faithful, and essentially surjective (every object of D is isomorphic to some F(C)). Skeleta, opposite categories of finite sets, and many duality theorems (Stone duality, Pontryagin duality) are equivalences.
Verify that the inclusion of the full subcategory of finite sets {∅, {1}, {1,2}, {1,2,3}, ...} into FinSet is an equivalence, even though it is not an isomorphism. Check full faithfulness and essential surjectivity explicitly. Then compare with the non-equivalence of Set and Grp to understand why all three conditions are necessary.
When you studied natural transformations and isomorphisms in categories, you learned that two objects in a category can be 'the same' in the categorical sense if they are isomorphic — even if they are not literally equal. Equivalence of categories extends this idea to entire categories: two categories C and D are equivalent if there exist functors F: C → D and G: D → C such that GF is naturally isomorphic to the identity on C, and FG is naturally isomorphic to the identity on D. Crucially, GF does not have to *equal* the identity — it only needs to be naturally isomorphic to it, meaning each component GF(A) ≅ A naturally and coherently.
This distinction from isomorphism of categories is profound. A true isomorphism would require GF = Id_C — going from C to D and back gives you the *same* objects and morphisms, not merely isomorphic ones. Equivalence allows F and G to reshuffle objects as long as the reshuffled objects are isomorphic to the originals. This makes equivalence the correct notion of 'sameness' for categories, since all categorical properties (limits, colimits, adjunctions) are invariant under natural isomorphism — not just under equality.
The three-condition characterization gives a practical test: a functor F: C → D is an equivalence if and only if it is (1) full — every morphism in D between objects in the image of F lifts back to C; (2) faithful — F reflects distinctions, so distinct morphisms in C remain distinct in D; and (3) essentially surjective — every object of D is isomorphic to some F(c). All three conditions are necessary. Faithful ensures F doesn't collapse structure; full ensures no new structure appears in D that wasn't in C; essentially surjective ensures every object of D is covered up to isomorphism.
A concrete example makes this vivid: consider the category FinSet of all finite sets and functions, and the full subcategory S consisting of exactly the sets {1}, {1,2}, {1,2,3}, ... — one representative of each finite cardinality. The inclusion S → FinSet is not surjective on objects (there are many 2-element sets, but only {1,2} is in S). But every finite set is isomorphic to exactly one representative, so the inclusion is essentially surjective. Since S is a full subcategory, the functor is also full and faithful. Therefore the inclusion is an equivalence — S and FinSet are 'the same category' categorically, even though they have very different object sets.
This is why equivalence is the standard notion used in duality theorems across mathematics. Stone duality, Pontryagin duality, and Morita equivalence are all equivalences of categories that translate between seemingly different mathematical worlds while preserving all relevant categorical structure. Recognizing an equivalence means recognizing that two areas of mathematics are, at a deep structural level, the same.