\begin{definition}[Definition:Set Product/Family of Sets]
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.
Let $P$ be a set.
Let $\family {\phi_i}_{i \mathop \in I}$ be an indexed family of mappings $\phi_i: P \to S_i$ for all $i \in I$ such that:
:For all sets $X$ and all indexed families $\family {f_i}_{i \mathop \in I}$ of mappings $f_i: X \to S_i$ there exists a unique mapping $h: X \to P$ such that:
::$\forall i \in I: \phi_i \circ h = f_i$
:that is, such that for all $i \in I$:
::$\begin{xy}\xymatrix@+1em@L+3px{
 X
  \ar@{-->}[d]_*+{h}
  \ar[dr]^*+{f_i}
\\
 P
  \ar[r]_*{\phi_i}
&
 S_i
}\end{xy}$
:is a commutative diagram.
Then $P$, together with the family of mappings $\family {\phi_i}_{i \mathop \in I}$, is called '''a product of (the family) $\family {S_i}_{i \mathop \in I}$'''.
This '''product of $\family {S_i}_{i \mathop \in I}$''' can be denoted $\struct {P, \family {\phi_i}_{i \mathop \in I} }$.
\end{definition}