Pushout witness for failure of pointwise surjectivity ($Y{⨿}_{X}Y$)

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 $Y\rightrightarrows Y{⨿}_{X}Y$ 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.