Sequent systems for classical anti intuitionistic logic and natural deduction systems for these logics are characterized by two important theorems. Sequent systems are characterized by cut- elimination theorems, and natural deduction systems by normalization theorems. In this paper, by means of multicategories and the typed $\lambda$-calculus we exhibit some similarities and differences between cut elimination and normalization. We consider the sequent system and the natural deduction system for intuitionistic propositional logic. We define a multicategory corresponding to the sequent system. On the other hand, a typed $\lambda$-calculus corresponds to the natural-deduction system. We show how to form a typed $\lambda$-calculus out of a multicategory and vice-versa. In some kinds of multicategory, some equations necessary for cut elimination, are not. necessary for normalization.