A map $ f{\colon}\linebreak[0] A \to B $ in a category $ \mathscr{A} $ is an isomorphism if there exists a map $ g{\colon}\linebreak[0] B \to A $ in $ \mathscr{A} $ such that $ gf = 1_A $ and $ fg = 1_B $ . In the situation of Definition X, we call $ g $ the inverse of $ f $ and write $ g = f^{ - 1} $ . (The word `the’ is justified by Exercise X.)