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In proving the Fermionic formulae, combinatorial bijection called theKerov--Kirillov--Reshetikhin (KKR) bijection plays the central role. It is abijection between the set of highest paths and the set of riggedconfigurations. In this paper, we give a proof of crystal theoreticreformulation of the KKR bijection. It is the main claim of Part I(math.QA/0601630) written by A. Kuniba, M. Okado, T. Takagi, Y. Yamada, and theauthor. The proof is given by introducing a structure of affine combinatorial$R$ matrices on rigged configurations.Comment: 45 pages, version for publication. Introduction revised, more  explanations added to the main tex

Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection II. Proof for $\mathfrak{sl}_{n}$ case