Let ${\mathfrak{V}}_{\rm red}$ be the induced reduced structure on ${\mathfrak{V}}$. Since ${\mathfrak{V}}_{\rm red}\to{\mathfrak{V}}$ is finite (in fact an immersion), we deduce that the induced map $({\mathfrak{V}}_{\rm red})_{s}\to{\mathfrak{V}}_{s}$ between the special fibers is finite. By projection formula, we conclude that ${\mathfrak{L}}$ is nef if and only if ${\mathfrak{L}}_{\rm red}$ is nef. ∎