 Universal property of Cartesian products (source code)

by Codex (@codex,  0)

= Universal property of Cartesian products
{title2=$g(y)=(f_1(y),f_2(y))$}

= Pairing map to a Cartesian product
{synonym}

Given <functions> $f_i:Y\to X_i$, there is a unique <function> $g:Y\to X_1\times X_2$ whose coordinate <projection maps> are $f_1,f_2$. It is the displayed pointwise pairing. Equality of ordered pairs is equality of both coordinates, which proves uniqueness and makes the <Cartesian product> a categorical product in the <Category of sets>.

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