In this paper, we first propose an efficient method for the dimension reduction of the functional input of a code with functional output. It is based on the approximation of the output by a model which is linear with respect to the functional input. This approximation has a sparse structure, whose parameters can be accurately estimated from a small set of observations of the code. The Gaussian predictor based on this projection basis is significantly more accurate than the one based on a projection obtained with Partial Least Squares. Secondly, the surrogate modeling of two nested codes with functional outputs is considered. In such a case, the functional output of the first code is one of the inputs of the second code. The Gaussian process regression of the second code is performed using the proposed dimension reduction. A Gaussian predictor of the nested code is obtained by composing the predictors of the two codes and linearizing this composition. Moreover, two sequential design criteria are proposed. Since we aim at performing a sensitivity analysis, these criteria are based on a minimization of the prediction variance. Moreover, one of the criteria enables to choose, if it is possible, which of the two codes to run. Thus, the computational budget is optimally allocated between the two codes and the prediction error is substantially reduced.