<p>In this paper, we focus on the functional sufficient dimension reduction (FSDR) problem, where the predictor is a function and the response is a vector. By projecting the multivariate response to univariate forms, we propose a projection averaging method for estimating the functional central subspace. The proposed method avoids multivariate smoothing of responses and can fully recover the central subspace under reasonable conditions. By employing functional sliced inverse regression, functional sliced average variance estimator, and functional cumulative slicing as the underlying univariate-response FSDR methods, we develop three specific methods. Furthermore, we propose a class of distance-based methods. We establish the convergence rates for these methods under certain smoothness assumptions. Simulation studies and a real data application demonstrate the superior performance of the proposed methods.</p>