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In Conceptual Mathematics 1st edition, p. 325-236, there is a sketch of a proof, but I can't carry out the complete proof.

"... This also follows from the appropriate universal mapping properties, which imply that the two composites satisfy properties that only the corresponding identity maps satisfy."

I can't figure this out.

Can you give me a clue?

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    @user5158: It would be useful if you included a bit more information of what they are proving; that is, provide enough context to know what they are doing. The way you've written the question, only the people who have access to the book *right now* and are willing to go look at it will be able to help you. If you can give enough context, I suspect I would be able to help you out. But right now, I cannot.2010-12-28
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    @user5158: But usually, this kind of argument boils down to this: functions $f$ and $g$, with $f$ going form an object $C$ to an object $D$, and $g$ going from $D$ to $C$, where both $C$ and $D$ have a certain uniqueness-universal-property relative to some diagram commuting. It is then a matter of checking that both the map $fg\colon D\to D$ *and* the identity "fit" into the commutative diagram, so that by uniqueness you have $fg=\mathrm{id}_D$; then one does the same with $gf\colon C\to C$, so that $gf=\mathrm{id}_C$. These two imply that $f=g^{-1}$ and that they are isomorphisms.2010-12-28

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