uniqueness of a two-sided inverse morphism

Let \(\mathcal{C}\) be a category and \(f: A \rightarrow B\) for $A,B ∈ \mathrm{Ob}(\mathcal{C})$d an isomorphism. Then there exists a unique inverse morphism \(f^{-1}\)

By assumption, there exists a left-inverse \(f_l^{-1}\) and a right-inverse \(f_r^{-1}\) Let \(f_l^{-1}\) be a left-inverse and \(f_r^{-1}\) a right-inverse: Then we conclude that

\begin{align*} f_l^{-1} =& f_l^{-1} \circ \mathrm{id}_{B} \\ =& f_{l}^{-1} \circ f \circ f_r^{-1} \\ =& \mathrm{id}_{A} \circ f_r^{-1} \\ =& f_r^{-1} \end{align*}