= Unions of subobjects are pushouts
{title2=$A\cup B=A\amalg_{A\cap B}B$}
In a <coherent category>, maps $A\to Y$ and $B\to Y$ agreeing on the intersection define a relation from $A\cup B$ to $Y$ by taking the union of their graphs. It is total because the two subobjects cover their union, and single-valued because the maps agree on overlap. <Total functional relations define morphisms>, giving a unique glued map. This is the <pushout> universal property without assuming a general supply of pushouts.
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