Fibrewise constant morphisms from an abelian scheme factor through the base

Fix $t\in T$ and choose an affine open $W=\mathrm{Spec}B\subseteq Z$ containing the image point of the fibre $A_t$; this is possible because the fibre image is a single point. The complement $Z\setminus W$ is closed, and since $f_T$ is proper by [F2] the set $u^{-1}(Z\setminus W)$ has closed image in $T$; by construction that image misses $t$. Choose an affine neighbourhood $V=\mathrm{Spec}R$ of $t$ disjoint from the image; then $f_T^{-1}(D)$ for $D=T\setminus V$ is closed in $A_T$ and disjoint from $A_V$, so $u{\mathrm{\mid }}_{A_V}$ lands in $W$. Thus over an affine neighbourhood of every point the map factors through an affine target.

On $A_V$ the structure morphism $f_V:A_V\to V$ is proper and the restriction $u_V:A_V\to W=\mathrm{Spec}B$ corresponds by [F3] to a ring map $B\to \mathrm{\Gamma }(A_V,{\mathcal{O}}_{A_V})$. The universal-sections lemma [F1] identifies $\mathrm{\Gamma }(A_V,{\mathcal{O}}_{A_V})=\mathrm{\Gamma }(V,{\mathcal{O}}_V)=R$, so this ring map factors through $R$ and defines a morphism $g_V:V\to W=\mathrm{Spec}B$ with $u{\mathrm{\mid }}_{A_V}=g_V\circ f_V$. Evaluating along the unit section gives $u{\mathrm{\mid }}_{A_V}\circ e_V=g_V$, so $u{\mathrm{\mid }}_{A_V}=u{\mathrm{\mid }}_{A_V}\circ e_V\circ f_V$.

The local factorizations of step 2.1 agree on overlaps: on $V_1\cap V_2$ both $g_{V_1}$ and $g_{V_2}$ equal $u\circ e$ evaluated there, because $f$ restricted to the unit section is an isomorphism onto the base; hence they glue to a morphism $g:T\to Z$ with $u=g\circ f_T$, and $g=u\circ e_T$ by the same evaluation. Therefore $u=u\circ e_T\circ f_T$. This controls nilpotents: the factorization is an identity of morphisms, not merely of geometric points.