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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $A$, $B$ be objects of $\\mathcal{C}$.\nA *morphism* from $A$ to $B$ is a member $f$ of $\\Hom_{\\mathcal{C}}(A, B)$, written $f \\colon A \\to B$.\nThe object $A$ is its domain and the object $B$ is its codomain.\nSpecial kinds of morphisms are defined as follows:\n\n1. $f \\colon A \\to B$ is an *isomorphism* if there is $g \\colon B \\to A$ with $g \\circ f = 1_A$ and $f \\circ g = 1_B$;\n2. $f$ is an *endomorphism* if $A = B$;\n3. $f$ is an *automorphism* if it is both an endomorphism and an isomorphism.\n\n## Remarks\n\nThe morphism $g$ in clause 1 is unique: if $g'$ also works, then $g' = g' \\circ (f \\circ g) = (g' \\circ f) \\circ g = g$.\nIt is called the inverse $f^{-1}$, and $A$ and $B$ are then *isomorphic*, written $A \\cong B$.\nExample: in the category of groups, the morphisms are the [[group homomorphisms|0JYTG5K9Y4]].\nThe isomorphisms are the bijective homomorphisms, since the inverse map of a bijective homomorphism is again a homomorphism.\nNon-example: in the category of topological spaces with [[continuous maps|0VGFHTM75Y]], let $X$ be $[0, 1]$ with the discrete topology and $Y$ be $[0, 1]$ with the usual topology.\nThe identity map $X \\to Y$ is a continuous bijection, but its inverse $Y \\to X$ is not continuous, so it is not an isomorphism.\nThe endomorphisms of $A$ form a monoid $\\End(A)$ under composition, and the automorphisms form a group $\\Aut(A)$.\n",
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