A Function $f:X \to Y$ that is surjective and injective

A necessary and sufficient condition is that there exists a function $f^{-1}$ such that $f \circ f^{-1} = id_X$ (this guarantees it is injective) and $f^{-1} \circ f = id_Y$ (which guarantees it is surjective), where $id_A$ is the Identitiy map from $A$ to itself.