Isomorphism (In a Category)
Let \(\mathscr{C}\) be a category , and let \(a\) and \(b\) be object in \(\mathscr{C}\). We call a morphism \(f:a \rightarrow b\) an isomorphism between \(a\) and \(b\) provided there exists a morphism \(g:b \rightarrow a\) such that:
\[ g \circ f=id_a \text{ and } f \circ g=id_b \]If such an isomorphism exists between \(a\) and \(b\), we call \(a\) and \(b\) isomorphic.