 Fiber product of schemes

The fibre product $X{\times }_{S}Y$ represents pairs of morphisms to $X$ and $Y$ with equal composites to $S$. Affine-locally,

The fiber product of schemes is a fundamental construction in algebraic geometry, analogous to the notion of the fiber product in category theory. It allows us to "pull back" schemes along morphisms, producing a new scheme that encodes information from each of the original schemes.