# Categorical product

From Cattheory

## Definition

Suppose is a category and . A **categorical product**, or simply **product**, of and is an object along with morphisms (called *projection maps*) , such that the following holds:

For any object and morphisms , there is a *unique* morphism such that and .

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.