As an example of (B), I'd mention Connes's classification of injective (type $II_{1}$) factors from the theory of von Neumann algebras. In this case, many apparently disparate constructions turn out to give a single object.
Classification of injective factors. Cases II1$II_1$,$ $II_\infty $II_\infty$,$ $III_\lambda $III_\lambda$,$ $\lambda \not=1$ $\lambda \not=1$. Ann. of Math. (2) 104 (1976), no. 1, 73-115.