  We prove that every homomorphism $\mathcal{O}^E_\zeta\to\mathcal{O}^F_\zeta$,
with $E$ and $F$ Banach spaces and $\zeta\in\mathbb{C}^m$, is induced by a
$\mathop{\mathrm{Hom}}(E,F)$-valued holomorphic germ, provided that $1\leq
m<\infty$. A similar structure theorem is obtained for the homomorphisms of
type $\mathcal{O}^E_\zeta\to\mathcal{S}_\zeta$, where $\mathcal{S}_\zeta$ is a
stalk of a coherent sheaf of positive $\mathfrak{m}_\zeta$-depth. We later
extend these results to sheaf homomorphisms, obtaining a condition on coherent
sheaves which guarantees the sheaf to be equipped with a unique analytic
structure in the sense of Lempert-Patyi.
