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This paper proposes a new nonlinear approach for additive functional regression with functional response based on kernel methods along with some slight reformulation and implementation of the linear regression and the spectral additive model. The latter methods have in common that the covariates and the response are represented in a basis and so, can only be applied when the response and the covariates belong to a Hilbert space, while the proposed method only uses the distances among data and thus can be applied to those situations where any of the covariates or the response is not Hilbert, typically normed or even metric spaces with a real vector structure. A comparison of these methods with other procedures readily available in R is preformed in a simulation study and in real datasets showing the results of the advantages of the nonlinear proposals and the small loss of efficiency when the simulation scenario is truly linear. The comparison is done in the Hilbert case as it is the only scenario where all the procedures can be compared. Finally, the supplementary material provides a visualization tool for checking the linearity of the relationship between a single covariate and the response, another real data example, and a link to a GitHub repository where the code and data are available.} %and an example considering that the response is not Hilbertian.