Yoneda embedding detects split epimorphisms ($y(f)\text{ epic}⟺f\text{ split epic}$)

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.