# Pullback over monomorphism

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

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 $f : Y \to Z$, $g : W \to X$, $h : X \to Z$.