The product of a family of objects of a category is an object , together with a family of morphisms such that for every object and every family of morphisms there is a unique morphism such that
for all . The product is unique up to isomorphisms.
In the category of sets, the product is the Cartesian product, and in the category of groups it is the group direct product. In both cases, , and is the projection onto the th factor.
This entry contributed by Margherita Barile
More things to try:
Barile, Margherita. "Category Product." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CategoryProduct.html
of objects of a category
is an object 
,
together with a family of morphisms 
such that for
every object 
and every family of morphisms 
there is a unique morphism 
such that
.
The product is unique up to isomorphisms.
, and 
is the projection onto the 
th factor.
