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How can we see that a natural transformation $\alpha:F\to G$ between functors $F,G:C\to D$ is a natural isomorphism iff for each object $c$ in the category $C$, $\alpha_c$ is an isomorphism in $D.$

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up vote 6 down vote accepted

I assume that $F,G : C \to D$ are functors and $\alpha : F \to G$ is a natural transformation. You want to show that

(1) $\alpha$ is an isomorphism.

(2) each $\alpha_c : F(c) \to G(c)$ is an isomorphism ($c \in C$).

are equivalent. Well, (1) => (2) is very easy (use the definitions and nothing else). For (2) => (1), show that $\alpha^{-1} : G \to F$ defined by $(\alpha^{-1})_c := (\alpha_c)^{-1}$ is a natural transformation, which is inverse to $\alpha$.

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To prove this can we take opposite categories? Actually I an the bigenner in this subject, can you explain me a bit more. How do we find inverses such that diagram commutes. Thanks. – smaz Dec 1 '13 at 15:34
I won't explain more, since now it is your task to fill in the details. It's not much, you only have to play around with the definitions. No opposite categories. – Martin Brandenburg Dec 1 '13 at 18:31

Often, the easiest way to show something is an isomorphism is to find its inverse.

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