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In this paper we start by pointing out that Yoneda's notion of a regular span S : X → A × B can be interpreted as a special kind of morphism, that we call fiberwise opfibration , in the 2-category Fib (A). We study the relationship between these notions and those of internal opfibration and two-sided fibration. This fibrational point of view makes it possible to interpret Yoneda's Classification Theorem given in his 1960 paper as the result of a canonical factorization, and to extend it to a non-symmetric situation, where the fibration given by the product projection P r 0 : A × B → A is replaced by any split fibration over A. This new setting allows us to transfer Yoneda's theory of extensions to the non-additive analog given by crossed extensions for the cases of groups and other algebraic structures. [ABSTRACT FROM AUTHOR]