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Let $X,Y,Z$ and $W$ sets, $f:X \to Y$, $g:Y \to Z$ and $h:Z \to W$ functions. If $g \circ f$ and $h \circ g$ are bijective, then $f,g$ and $h$ are bijective.

My attemp of proof goes as follows: As $g \circ f$ and $h \circ g$ are both injective and surjective, then $f$ and $g$ are injective, $g$ and $h$ are surjective. So $g$ is bijective. How do I prove that $f$ is surjetive and $h$ is injective in order to show that $f$ and $h$ are also bijective?

Thanks.

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As you have already noted the information given shows that $g$ is bijective and hence so is $g^{-1}$. Since a composition of bijections is a bijection we have that $f = g^{-1} \circ g \circ f$ is a bijection and similarly for $h$.

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Because $g \circ f$ is bijective and $g^{-1}$ exists and is bijective, $g^{-1} \circ (g \circ f)$ is also bijective.

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