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Let $W\rightarrow X$ be a monomorphism. Is it true that $Y \times_Z W = W \times_X (Y\times_Z X)$, provided that $Y\times_Z X$ exists? It is not hard to get the map in the $\leftarrow$ direction, but how do I use the fact that $W\rightarrow X$ is mono to get the map in the other direction?

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You don't need the monomorphism hypothesis. This is just the pullback pasting lemma. –  Zhen Lin Jan 30 at 9:48
    
ncatlab.org/nlab/show/pullback –  a.r. Jan 30 at 10:00
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As already remarked by Zhen Lin, this always holds. Using the Yoneda-Lemma, we may reduce to the case of sets. Then we have a canonical bijection $$Y \times_Z W \to W \times_X (Y \times_Z X), (y,w) \mapsto (w,(y,g(w)))$$ where $f : Y \to Z$, $g : W \to X$, $h : X \to Z$.

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Cool, I never realized I could use Yoneda to generalize things that are obvious in Sets. Do you have a word to say of when it doesn't work to use Yoneda Lemma to prove an intuition I have of something that should work on a category because it works on Sets? –  Rodrigo Jan 30 at 23:00
    
Well I have quite some intuition for these things, but this came from many examples. I'm not sure how to describe it without saying something wrong ... better find your own intuition :) –  Martin Brandenburg Jan 30 at 23:25
    
Ok, sure! But if it ever occurs to you an example of something that is true for sets and not in general, and where you don't get what you expect if you use Yoneda Lemma to translate from one to the other, do inform me :) –  Rodrigo Jan 30 at 23:57
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Sure, for example that monomorphisms are stable under pushouts. This fails, for example, in the category of rings, and also in the category of groups, but it is true in the category of sets and the category of topological spaces. The reason is simply that we "want" to exchange limits with colimits here, but we cannot mix the covariant with the contravariant Yoneda embedding in order to say something about that. In general, commutation of limits with colimits is very delicate! –  Martin Brandenburg Jan 30 at 23:58
    
Great example! I'll add the link of another question here just for future reference of how one goes back-and-forth using Yoneda math.stackexchange.com/questions/107849/… –  Rodrigo Jan 31 at 0:49
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