From Copying to Corelations via Ancestry Partitions

We study the free PROP $\mathrm{S}\mathrm{y}\mathrm{n}(\delta )$ on a single binary generator $\delta :1\to 2$. The ancestry functor $\mathrm{\Pi }:\mathrm{S}\mathrm{y}\mathrm{n}(\delta )\to \mathrm{F}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}$, defined by connected components of the underlying undirected string diagram, has image the sub-PROP ${\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}}^{\circ }$ of finite corelations whose equivalence classes contain exactly one input and at least one output. The induced quotient [ $\mathrm{A}\mathrm{n}\mathrm{c}\mathrm{Q}$:=$\mathrm{S}\mathrm{y}\mathrm{n}$(δ)/\ker(Π) ] is equivalent as a PROP to $\mathrm{C}\mathrm{o}\mathrm{c}\mathrm{o}\mathrm{m}$, the PROP for non-counital cocommutative comonoids. We then locate this primitive construction inside the standard cospan/corelation framework: $\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(\mathcal{B})$ realizes pushout-style gluing as a free hypergraph category; $\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{S}\mathrm{e}\mathrm{t})$ collapses under jointly epic corestriction to $\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}$, the PROP for extraspecial commutative Frobenius monoids; and the Yoneda envelope [ \mathcal W=$\mathrm{F}\mathrm{u}\mathrm{n}$($\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{l}$^{op},$\mathrm{S}\mathrm{p}\mathrm{c}$) ] is a presheaf $\mathrm{\infty }$-topos carrying the standard subobject, modality, and monotone fixed-point apparatus. The PROP-level identification $\mathrm{A}\mathrm{n}\mathrm{c}\mathrm{Q}\simeq \mathrm{C}\mathrm{o}\mathrm{c}\mathrm{o}\mathrm{m}$ is the only result claimed as new; the remaining material is organizational and reduces explicitly to cited classical results.