 0 \rightarrow \mathbb{H}^0(B',L_{(f',i')}^\vee) \rightarrow \mathbb{H}^0(B, L_{(f,i)}^\vee) \rightarrow 0 \end{aligned}$$ So the canonical map from the space of first order deformations of $\zeta'$ to the space of first order deformations of $\zeta$ has both $0$-dimensional kernel and cokernel, 