Let $p:X\to Y$ be a proper holomorphic submersion between complex manifolds with compact fibres $X_y:=p^{-1}(y)$, and let $(L,h)$ be a smooth Hermitian holomorphic line bundle on $X$ with semipositive Chern curvature $i\Theta_h(L)\ge 0$. Suppose that \begin{align*} E:=p_*(K_{X/Y}\otimes L) \end{align*} is locally free, so that $E\to Y$ is a holomorphic vector bundle, and equip $E$ with the natural fibrewise $L^2$ metric \begin{align*} \langle u,v\rangle_y=c_n\int_{X_y}u\wedge\overline v\,h, \qquad c_n=i^{n^2}, \end{align*} for $u,v\in H^0(X_y,K_{X_y}\otimes L|_{X_y})$. Then the Chern curvature of $(E,\langle\cdot,\cdot\rangle)$ is Nakano semipositive.