nLab
inverse function

Given sets $A$ and $B$ and a function $f:A\to B$, the inverse function of $f$ (if it exists) if the function $f^{-1}:B\to A$ such that both composite function?s $f\circ f^{-1}$ and $f^{-1}\circ f$ are identity functions. Note that $f$ has an inverse function if and only if $f$ is a bijection, in which case this inverse function is unique.

Inverse functions are inverse morphisms in the category Set of sets.

More generally, in any concrete category, the inverse of any isomorphism is given by the inverse of the corresponding function between underlying sets.