\begin{definition}[Definition:Coproduct]
Let $\mathbf C$ be a metacategory.
Let $A$ and $B$ be objects of $\mathbf C$.
A '''coproduct diagram''' for $A$ and $B$ comprises an object $P$ and morphisms $i_1: A \to P$, $i_2: B \to P$:
::$\begin{xy}
<-4em,0em>*+{A} = "A",
<0em,0em>*+{P}  = "P",
<4em,0em>*+{B}  = "B",
"A";"P" **@{-} ?>*@{>} ?*!/^.8em/{i_1},
"B";"P" **@{-} ?>*@{>} ?*!/_.8em/{i_2},
\end{xy}$
subjected to the following universal mapping property:
:For any object $X$ and morphisms $x_1, x_2$ like so:
::$\begin{xy}
<-4em,0em>*+{A} = "A",
<0em,0em>*+{X}  = "X",
<4em,0em>*+{B}  = "B",
"A";"X" **@{-} ?>*@{>} ?*!/^.8em/{x_1},
"B";"X" **@{-} ?>*@{>} ?*!/_.8em/{x_2},
\end{xy}$
:there is a unique morphism $u: P \to X$ such that:
::$\begin{xy}
<0em,5em>*+{X}  = "X",
<-5em,0em>*+{A} = "A",
<0em,0em>*+{P}  = "P",
<5em,0em>*+{B}  = "B",
"A";"X" **@{-}  ?>*@{>} ?*!/^.8em/{x_1},
"B";"X" **@{-}  ?>*@{>} ?*!/_.8em/{x_2},
"P";"X" **@{--} ?>*@{>} ?*!/_.6em/{u},
"A";"P" **@{-}  ?>*@{>} ?*!/_.8em/{i_1},
"B";"P" **@{-}  ?>*@{>} ?*!/^.8em/{i_2},
\end{xy}$
:is a commutative diagram.
That is:
:$x_1 = u \circ i_i$ and $x_2 = u \circ i_2$
In this situation, $P$ is called a '''coproduct of $A$ and $B$''' and may be denoted $A + B$.
We generally write $\sqbrk {x_1, x_2}$ for the unique morphism $u$ determined by above diagram.
The morphisms $i_1$ and $i_2$ are often taken to be implicit.
They are called '''injections'''; if necessary, $i_1$ can be called the '''first injection''' and $i_2$ the '''second injection'''.
{{expand|the injection definition may merit its own, separate page}}
{{definition wanted|general definition}}
\end{definition}