The difference of two connections is an endomorphism valued one form

Subtracting the local formulas cancels $du$: for $s=eu$, the difference is $e({\omega }_1(X)-{\omega }_0(X))u$. At a point it depends only on $X(p)$ and $u(p)$, not their derivatives. Thus the matrix difference gives a smooth linear map $T_{p}M\to \mathrm{End}(E_p)$ in each chart, including zero inputs.

Under $e^{\mathrm{\prime }}=eA$, the two $A^{-1}dA$ terms cancel and the matrix difference becomes $A^{-1}({\omega }_1-{\omega }_0)A$. This is exactly change of basis for the same fibre endomorphism. These local maps agree intrinsically and define the claimed smooth Hom section. Every fibre vector is a value of a local constant-coefficient section; the equality for these sections determines the endomorphism uniquely. Equal connections give $B=0$; rank one gives scalar one-forms, rank zero gives the unique zero map, and an empty base imposes no values. This is a comparison of supplied connections and uses no existence theorem or choice axiom.