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Suppose $f$ and $g$ are injective maps such that $h \circ f = g $. What are the requirements on $h$?

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Denoting by $Im(f)$ the image of the map $f$, the only requirement you need on $h$ is that $h$ is defined at least on $Im(f)$ and injective on this set.

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Well, $h$ must be defined as $g\circ f^{-1}$ on the image of $f$. Elsewhere, it may be anything.

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