  We derive sufficient conditions for the surjectivity of the Cauchy-Riemann
operator $\overline{\partial}$ between spaces of weighted smooth
Fr\'echet-valued functions. This is done by establishing an analog of
H\"ormander's theorem on the solvability of the inhomogeneous Cauchy-Riemann
equation in a space of smooth $\mathbb{C}$-valued functions whose topologyis
given by a whole family of weights. Our proof relies on a weakened variant of
weak reducibility of the corresponding subspace of holomorphic functions in
combination with the Mittag-Leffler procedure. Using tensor products, we deduce
the corresponding result on the solvability of the inhomogeneous Cauchy-Riemann
equation for Fr\'echet-valued functions.
