Binary Pullback

A pullback of two morphisms $f:{{X}\to{Z}}$, $g:{{Y}\to{Z}}$ with common codomain in a Category $\mathcal{C}$ is an object $X\times_{Z}Y$ equipped with a pair of projections $\pi_{1}:X\times_{Z}Y\to X$, $\pi_{2}:X\times_{Z}Y\to Y$ such that

• $f\circ\pi_{1}=g\circ\pi_{2}$
• For any other $P$ equipped with maps $p_{1}:P\to X$, $p_{2}:P\to Y$ with $f\circ p_{1}=g\circ p_{2}$, there exists a unique universal map $u:P\to X\times_{Z}Y$ such that $\pi_{1}\circ u=p_{1}$ and $\pi_{2}\circ u=p_{2}$.

Properties

• A pullback of two morphisms with codomain $Z$ is a Binary Cartesian Product in a Slice Category $\mathcal{C}/Z$.
• A pullback is a Limit of a Cospan.