 Solution

Let $h:B\to A$ be a two-sided inverse function of $f$. If $f(a)=f(a^{\prime })$, apply $h$ and use $h\circ f={\mathrm{id}}_A$ to obtain $a=a^{\prime }$, proving injectivity. For any $b\in B$, the other identity gives $b=f(h(b))$, proving surjectivity. Therefore $f$ is a bijection. Both inverse identities matter: a one-sided inverse alone would establish only the corresponding one of these two properties.