If two compositions are bijective, then all functions involved are bijective?

Question

Given functions $f:A\to B$, $g:B\to C,$ and $h:C\to D.$ Provided $g\circ f$ and $h\circ g$ are bijective, prove each of the functions $f$, $g$, and $h$ is bijective.

Answer 1

Suggestion: show first that for functions $f: A \rightarrow B$ and $g: B \rightarrow C$, if $g \circ f$ is injective, then $f$ is injective, and if $g \circ f$ is surjective, then $g$ is surjective. Then use this to answer the question.