  The universal enveloping algebra functor between Leibniz and associative
algebras defined by Loday and Pirashvili is extended to crossed modules. We
prove that the universal enveloping crossed module of algebras of a crossed
module of Leibniz algebras is its natural generalization. Then we construct an
isomorphism between the category of representations of a Leibniz crossed module
and the category of left modules over its universal enveloping crossed module
of algebras. Our approach is particularly interesting since the actor in the
category of Leibniz crossed modules does not exist in general, so the technique
used in the proof for the Lie case cannot be applied. Finally we move on to the
framework of the Loday-Pirashvili category $\mathcal{LM}$ in order to
comprehend this universal enveloping crossed module in terms of the Lie crossed
modules case.
