In the present work we extend to crossed modules the classical adjunctionbetween the Liezation functor Liea : As Lie, which makes every associative algebraA into a Lie algebra via the bracket a,b ab ba , for all a,b A, and U : Lie As, which assigns to every Lie algebra p its universal enveloping algebra U( p ).Likewise, we construct a 2 dimensional generalization of the adjunction between thefunctor Lb : Di Lb, which assigns to every dialgebra D the Leibniz bracket givenby 1 2 1 d ,d d ┤ 2 2 d d ├ 1 d , for all d d D 1 2 , , and Ud : Lb Di, the universalenveloping dialgebra functor. Additionally, we assemble all the resulting squares ofcategories and functors in four parallelepipeds, for which, in every face, the inner andouter squares are commutative or commute up to isomorphism.Since our second generalization involves crossed modules of dialgebras, we givean adequate definition for them, based on the more general notion of crossed modules incategories of interest. Furthermore, we define the concept of strict 2 dialgebra, byanalogy to the notion of strict associative 2 algebra. We prove that the categories ofcrossed modules of dialgebras and strict 2 dialgebras are equivalent.Additionally, we construct the dialgebra of tetramultipliers, which happens to bethe actor in the category of dialgebras under certain conditions. Besides, given a Leibnizcrossed module, we construct a general actor crossed modules, which is the actor insome particular cases.