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?

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