Categorical product

Definition

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

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

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.