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  "body": "\n## Definition\n\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be a [[functor|28NQ0KRHD3]].\nFor objects $A$, $B$ of $\\mathcal{C}$, let $F_{A, B} \\colon \\Hom_{\\mathcal{C}}(A, B) \\to \\Hom_{\\mathcal{D}}(F(A), F(B))$ be the map $f \\mapsto F(f)$.\n\n1. $F$ is *faithful* if $F_{A, B}$ is [[injective|09NWETVTNY]] for all objects $A$, $B$.\n2. $F$ is *full* if $F_{A, B}$ is [[surjective|01RVCMSN5N]] for all objects $A$, $B$.\n3. $F$ is *fully faithful* if $F_{A, B}$ is bijective for all objects $A$, $B$.\n\n## Remarks\n\nA faithful functor may still identify two morphisms with different domains or codomains, and it may send different objects to the same object.\nExample: the forgetful functor from [[groups|G5S1M9]] to sets is faithful, since two [[homomorphisms|0JYTG5K9Y4]] with the same underlying map are equal.\nIt is not full: the constant map $\\Z \\to \\Z$ with value $1$ is a function between the underlying sets but not a homomorphism.\nThe inclusion of a [[full subcategory|V3BFGKBJF7]] is fully faithful.\nNon-example: the functor from groups to the category with one object and one morphism is not faithful.\nThe two homomorphisms $\\Z \\to \\Z$ given by $x \\mapsto 0$ and $x \\mapsto x$ are sent to the same morphism.\nA fully faithful functor reflects isomorphisms: if $F(f)$ is an isomorphism, then so is $f$.\n",
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