left inverse of an isomorphism as right inverse

Let \(\mathcal{C}\) be a category, \(f: A \rightarrow B\) an isomorphism and \(g: B \rightarrow A\) another morphism TFAE:

1. \(g\) is a left-inverse
2. \(g\) is a right-inverse
3. \(g\) is the inverse and hence an iso

By assumption, there exists a both sided inverse \(g'\). Then

\begin{align*} g =& g \circ \mathrm{id} \\ =& g \circ f \circ g' \\ =& (g \circ f) \circ g' \\ =& \mathrm{id} \circ g' =& g' \end{align*}

the other implication follows analogously or dually in \(\mathcal{C}^{\mathrm{op}}\)