fibre product

Let $S$ be a scheme, and let $i:X\longrightarrow S$ and $j:Y\longrightarrow S$ be schemes over $S$ . A fibre product of $X$ and $Y$ over $S$ is a scheme $X\times_{S}Y$ together with morphisms

	$\displaystyle p:X\times_{S}Y\longrightarrow X$
	$\displaystyle q:X\times_{S}Y\longrightarrow Y$

such that given any scheme $Z$ with morphisms

	$\displaystyle x:Z\longrightarrow X$
	$\displaystyle y:Z\longrightarrow Y$

where $i\circ x=j\circ y$ , there exists a unique morphism

(x,y):Z\longrightarrow X\times_{S}Y

making the diagram