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  "body": "\n## Statement\n\nAssume the axiom of choice, applied to the classes of objects involved.\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be a [[functor|28NQ0KRHD3]].\nThen $F$ is an [[equivalence of categories|K7A04ZBD92]] if and only if $F$ is [[fully faithful|GRG8WEBSH5]] and [[essentially surjective|6ERBSBVF14]].\n\n## Remarks\n\nSuppose first that $F$ has a quasi-inverse $G$ with [[natural isomorphisms|KYCPCC57FT]] $\\eta \\colon 1 \\Rightarrow GF$ and $\\varepsilon \\colon FG \\Rightarrow 1$.\nThen $\\varepsilon_D \\colon F(G(D)) \\cong D$ gives essential surjectivity, and naturality $\\eta_B \\circ f = G(F(f)) \\circ \\eta_A$ shows that $F$ is faithful, and likewise $G$.\nFor $h \\colon F(A) \\to F(B)$, the morphism $f = \\eta_B^{-1} \\circ G(h) \\circ \\eta_A$ satisfies $G(F(f)) = G(h)$, so $F(f) = h$ because $G$ is faithful; this direction needs no choice.\nConversely, choose for each object $D$ of $\\mathcal{D}$ an object $G(D)$ of $\\mathcal{C}$ and an isomorphism $\\varepsilon_D \\colon F(G(D)) \\to D$.\nFor $g \\colon D \\to D'$, full faithfulness gives a unique $G(g)$ with $F(G(g)) = \\varepsilon_{D'}^{-1} \\circ g \\circ \\varepsilon_D$, which makes $G$ a functor and $\\varepsilon$ natural.\nThe unit $\\eta_A$ is the unique morphism with $F(\\eta_A) = \\varepsilon_{F(A)}^{-1}$, and it is an isomorphism because fully faithful functors reflect isomorphisms.\nExample at work: let $K$ be a field, and include the full subcategory of spaces $K^n$ into all finite-dimensional vector spaces over $K$.\nThis inclusion is fully faithful and essentially surjective, so it is an equivalence.\nWithout choice, the converse can fail, since a quasi-inverse requires picking one preimage up to isomorphism for every object.\nIn fact the converse for all functors between small categories is equivalent to the axiom of choice.\n\n## Explanations\n\n### L1 human\n\nTo check that two categories are \"the same up to isomorphism\", one does not need to build a functor going back.\nIt is enough that the functor matches the arrows exactly and reaches every object up to isomorphism.\nThe return functor can then be built by picking, for each target object, one source object that lands near it.\n\n### L2 human\n\nThis criterion is how most equivalences are proved in practice, for instance that every category is equivalent to its skeleton.\nThe quasi-inverse constructed in the proof can be chosen so that $\\eta$ and $\\varepsilon$ satisfy the triangle identities, so every equivalence is part of an adjoint equivalence.\nProperties defined by universal properties, such as having products or being abelian, are invariant under equivalence, so the theorem lets one transfer them along any fully faithful essentially surjective functor.\n",
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