Suppose I am working in the ${\rm FinSet}$ category, and suppose a finite set $S$, say, containing a finite amount of character strings.
Consider now the slice category ${\rm C} = {\rm FinSet}/S$. Consider elements $a = (\{1\}, \{1 \rightarrow \text{"Alice"}\})$ and $b = (\{1\}, \{1 \rightarrow \text{"Bob"}\})$.
What categorical operation on two elements of ${\rm C}$ would find an element $r$ of ${\rm C}$ and two morphisms $i_1$ and $i_2$ with codomain $r$ such that:
- When given $a$ and $a$, $r$ is $a$, $i_1 = i_2 = {\rm Id}_{\{1\}}$
- When given $b$ and $b$, $r$ is $b$, $i_1 = i_2 = {\rm Id}_{\{1\}}$
- When given $a$ and $b$, $r$ is $(\{1,2\}, \{1 \rightarrow \text{"Alice"},2 \rightarrow \text{"Bob"}\})$, $i_1 = \{ 1 \rightarrow 1\}$ and $i_2 = \{ 1 \rightarrow 2\}$
I thought about the coproduct, but it seems that the former two conditions would yield a set with two elements, just like the third.