= Yoneda embedding detects split epimorphisms
{c}
{title2=$y(f)\text{ epic}\iff f\text{ split epic}$}
For $f:A\to B$, the <Yoneda embedding> sends $f$ to postcomposition $\mathcal C(-,A)\to\mathcal C(-,B)$. If this is epic in the <presheaf category>, its component at $B$ is surjective, so $1_B=fg$ for some $g:B\to A$. Conversely such a section makes the natural transformation split epic. Pointwise surjectivity follows directly by doubling the target presheaf along the image subpresheaf: the two inclusions agree on the image but differ on any element outside it.
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