RO-NMO 2017 Problem 104

Find the number of functions $A\stackrel{f}{\longrightarrow } A$ for which there exist two functions $A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A$ having the properties that $g\circ h =\text{id.}$ and $h\circ g=f,$ where $B$ and $A$ are two finite sets.