= Pushout witness for failure of pointwise surjectivity
{title2=$Y\amalg_XY$}
For a <natural transformation> $a:X\to Y$ of set-valued <functors>, form the <pushout> by taking two copies of each $Y(C)$ and identifying the two copies of every element in the image of $a_C$. <Naturality> makes all induced maps well-defined. If $y\in Y(C)$ is outside that image, its two copies remain distinct. Thus the two maps $Y\rightrightarrows Y\amalg_XY$ agree after $a$ but differ at $y$, proving $a$ is not an <epimorphism>. Conversely, componentwise <surjective functions> allow cancellation at every object, proving that $a$ is an epimorphism. This argument does not assume a natural choice of preimages.
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