Задача 12.

Show that the following properties of a map $f \mkern-2mu\colon\mkern-2mu X\to Y$ are equivalent 1) $f$ is bijective; 2) $f$ is surjective and injective; 3) $f$ has an inverse, i.e. there is a map $g\colon Y\to X$ such that $g\circ f=\mathrm{Id}_X$, $f\circ g=\mathrm{Id}_Y$, where $\mathrm{Id}_M\colon M\to M$, $m\mapsto m$ is the identity map on a set $M$.