= Product
{disambiguate=category theory}
{title2=$\prod$}
{wiki}
= Product in a category
{synonym}
= Products in a category
{synonym}
= Categorical product
{synonym}
A binary categorical product of <objects of a category> $A,B$ is an object $P$ with <categorical morphisms> $p_A:P\to A$ and $p_B:P\to B$. Its <universal property> is that, for every object $X$ and pair of arrows $f:X\to A$, $g:X\to B$, there is exactly one arrow $h:X\to P$ satisfying
$$
p_A\circ h=f,\qquad p_B\circ h=g.
$$
The arrows $p_A,p_B$ are called the product projections. Existence lets us combine two maps; uniqueness means there is no extra unspecified information in the combined map.
In the <Category of sets>, take $A=\{0,1\}$ and $B=\{a,b,c\}$. Their <Cartesian product> contains six <ordered pairs>, and the <projection maps> select the first and second coordinates. Let $X=\{u,v,w\}$, $f(u)=0$, $f(v)=1$, $f(w)=1$, and $g(u)=a$, $g(v)=b$, $g(w)=a$. The unique required map is \b[$h(u)=(0,a)$, $h(v)=(1,b)$, $h(w)=(1,a)$]. Reading either coordinate recovers the corresponding input map.
\Image[/categorical-product-factorization.png]
{title=Two maps into the factors determine exactly one map into their six-element Cartesian product}
In a <partially ordered set> viewed as a <category>, the projection arrows mean $P\leq A$ and $P\leq B$. The <universal property> says that every common <lower bound> $X$ satisfies $X\leq P$, so $P$ is the <greatest lower bound>. For <subsets> ordered by inclusion this is $A\cap B$, the <intersection>; it is a different construction from the <Cartesian product> in the <Category of sets> because the allowed arrows have changed.
The <universal property> also compares any two proposed products: each supplies a unique map to the other compatible with the projections. The two composites have the same projections as the corresponding <identity morphisms>, so uniqueness makes them identities. Thus a <categorical product> is determined up to a unique projection-preserving <isomorphism>, which explains why we can use whichever concrete construction is convenient.
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