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In the forgetful functor Wikipedia article I read that

"[Forgetful] Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set."

Can anyone give me an example of a forgetful functor that is not faithful?

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    $\begingroup$ The Wikipedia article gives examples for all three types of forgetful functors, including functors of the "third kind". $\endgroup$ – Dietrich Burde Jun 28 '15 at 12:52
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One example is the forgetful functor from Schemes to Sets. Given two fields $k_1, k_2$, there may be many different field homomorphisms $k_1 \to k_2$ which give rise to many different morphisms $\text{Spec }k_2 \to \text{Spec }k_1$. However, the underlying sets of both of these schemes are single points, so there is a unique map between them in the category Sets.

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  • $\begingroup$ Remark: There is a faithful functor from schemes (even ringed spaces) to sets, though. It is a good exercise to find it. $\endgroup$ – Martin Brandenburg Jun 28 '15 at 13:32
  • $\begingroup$ @MartinBrandenburg can I have a hint? $\endgroup$ – Tyche Jun 29 '15 at 14:40
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    $\begingroup$ Well, gather all pieces of the structure of a ringed space into one big set. $\endgroup$ – Martin Brandenburg Jun 29 '15 at 14:46
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If $\mathcal{C},\mathcal{D}$ are categories, then the projection functor $\mathcal{C} \times \mathcal{D} \to \mathcal{C}$ (which "forgets" the second coordinate) is not faithful (unless $\mathcal{D}$ is thin or $\mathcal{C}$ is empty).

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