\begin{definition}[Definition:Set Product]
Let $S$ and $T$ be sets.
Let $P$ be a set and let $\phi_1: P \to S$ and $\phi_2: P \to T$ be mappings such that:
:For all sets $X$ and all mappings $f_1: X \to S$ and $f_2: X \to T$ there exists a unique mapping $h: X \to P$ such that:
::$\phi_1 \circ h = f_1$
::$\phi_2 \circ h = f_2$
:that is, such that:
::$\begin{xy}\xymatrix@+1em@L+3px{
&
 X
  \ar[ld]_*+{f_1}
  \ar@{-->}[d]^*+{h}
  \ar[rd]^*+{f_2}
\\
 S
&
 P
  \ar[l]^*+{\phi_1}
  \ar[r]_*+{\phi_2}
&
 T
}\end{xy}$
:is a commutative diagram.
Then $P$, together with the mappings $\phi_1$ and $\phi_2$, is called '''a product of $S$ and $T$'''.
This '''product of $S$ and $T$''' can be denoted $\struct {P, \phi_1, \phi_2}$.
\end{definition}