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I will present a first-principles derivation of the AdS5/CFT4 T-system up to first non-trivial order in the large 't Hooft coupling expansion. The proof relies on the computation of quantum effects in the fusion of some special line operators, namely the transfer matrices. This computation is done in the pure spinor formalism for the superstring in AdS5xS5. I will also discuss the generalization of this computation to other integrable 2D CFTs that define string theory in AdS backgrounds.