Product

Let be a Category and A product of and is an Object together with Morphisms and with the universal property that for any with maps and there exists a unique such that we have a Commutative Diagram

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
& X \arrow[dddl, swap, "p_{1}"] 
\arrow[dd,"f"]
\arrow[dddr, "p_{2}"] \\
\\
& A\times B \arrow[dl, "\pi_{1}"]
\arrow[dr, swap, "\pi_{2}"] \\
A & & B
 
\end{tikzcd}
\end{document}

When is Locally Small, we can define a Functor defined by

If this is Representable, let be its Representation. i.e. is a Natural Isomorphism Note that is the Universal Element, consisting of where and and thus is the product of and .