\begin{definition}[Definition:Product (Category Theory)/General Definition]
Let $\mathbf C$ be a metacategory.
Let $\CC$ be any collection of objects of $\mathbf C$.
Let $\map {\mathbf {Dis} } \CC$ be the discrete category on $\CC$, considered as a subcategory of $\mathbf C$.
A '''product''' for $\CC$, denoted $\ds \prod \CC$, is a limit for the inclusion functor $D: \map {\mathbf {Dis} } \CC \to \mathbf C$, considered as a diagram.
For an object $C$ in $\CC$, the associated morphism $\ds \prod \CC \to C$ is denoted $\pr_C$ and called the '''projection on $C$'''.
The whole construction is pictured in the following commutative diagram:
::$\begin{xy}\xymatrix@R-.5em@L+3px{
& &
 A
  \ar@{-->}[dd]
  \ar[dddl]_*+{a_C}
  \ar[dddr]^*+{a_C'}
\\ \\ & &
 \ds \prod \CC
  \ar[dl]^*{\pr_C}
  \ar[dr]_*{\pr_{C'}}
\\
 \map {\mathbf {Dis} } \CC
&
 C
&
 \dots \quad \dots
&
 C'
}\end{xy}$
\end{definition}