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Isomorphism**

Given a category \(\mathcal{C}\), an arrow \(f:X\to Y\) in \(\mathcal{C}\) is called an isomorphism (or 'invertible') if there exists an arrow \(\tilde{f}:Y\to X\) such that \(f\circ \tilde{f}=1_Y\) and \(\tilde{f}\circ f=1_X\). If such an \(\tilde{f}\) exists then it is necessarily unique and generally written \(f^{-1}\), and \(X\) and \(Y\) are called 'isomorphic'. Isomorphism is an equivalence relation on the objects of \(\mathcal{C}\), and means that \(X\) and \(Y\) are essentially the same (from the point of view of the category).