K-morphisms from smooth models into abelian schemes extend uniquely

Restriction produces a well-defined map ${\mathrm{Hom}}_S(Z,A)\to {\mathrm{Hom}}_K(Z_K,A_K)$, injective because $A$ is separated over $S$ and $Z_K$ is schematically dense in the flat $S$-scheme $Z$; so it remains to show surjectivity.

For each height-one point $\xi$ of $Z$ lying over a closed point $s\in S$, [F1] gives a DVR ${\mathcal{O}}_{Z,\xi }$ whose fraction field is the function field of the component of $Z$ containing $\xi$. The restriction of $u_K$ gives a point of $A$ over that fraction field, and properness [F2] extends it to an $S$-morphism $\mathrm{Spec}{\mathcal{O}}_{Z,\xi }\to A$. By [F4] this map spreads to an open neighbourhood $W_{\xi }$ of $\xi$ in $Z$. Its restriction to $W_{\xi }\cap Z_K$ equals $u_K$, since they agree at the generic point and $A$ is separated. Independently, $u_K$ spreads to a morphism on an open neighbourhood $W_0\subseteq Z$ containing the whole generic fibre: work locally on a finite-type affine open of the Dedekind base, apply [F4] to its generic localization and the finitely presented smooth source and target, and then glue the resulting restrictions by separatedness. Put $V=W_0\cup {\bigcup }_{\xi }{W}_{\xi }$. This is an open subscheme containing the generic fibre and every vertical height-one point; every horizontal height-one point lies in the generic fibre. For each closed $s$, $V_s$ contains the generic points of all components of the smooth, hence reduced fibre $Z_s$, so $V$ is $S$-dense. The local extensions agree with each other and with $u_K$ on overlaps: the generic fibre is schematically dense in every open subscheme of the flat $Z$, and $A$ is separated. Thus they glue to a morphism $V\to A$, which represents an $S$-rational map defined at every height-one point.

The base $S$ is regular Noetherian and $Z$ is smooth over $S$, so Weil's extension theorem [F3] applies to the $S$-rational map represented in step 2.1. It extends uniquely to an $S$-morphism $Z\to A$ restricting to $u_K$. This proves surjectivity and hence the bijection. Uniqueness also follows from separatedness and schematic density of $Z_K$. The argument is applied componentwise; components of a smooth scheme over a Dedekind base are disjoint locally.