Categorical product

Definition

Suppose ${\mathcal {C}}$ is a category and $A_{1},A_{2}\in \operatorname {Ob} {\mathcal {C}}$. A categorical product, or simply product, of $A_{1}$ and $A_{2}$ is an object $C\in \operatorname {Ob} {\mathcal {C}}$ along with morphisms (called projection maps) $\pi _{1}:C\to A_{1},\pi _{2}:C\to A_{2}$, such that the following holds:

For any object $D\in {\mathcal {C}}$ and morphisms $f_{i}:D\to A_{i}$, there is a unique morphism $g:D\to C$ such that $\pi _{1}\circ g=f_{1}$ and $\pi _{2}\circ g=f_{2}$.

If a categorical product exists for two objects, then the categorical product is unique upto canonical isomorphism: for any two categorical products, there is a unique isomorphism between them that commutes with the projection maps.