  In this paper, first we construct two subcategories (using symmetric
representations and antisymmetric representations) of the category of relative
Rota-Baxter operators on Leibniz algebras, and establish the relations with the
categories of relative Rota-Baxter operators and relative averaging operators
on Lie algebras. Then we show that there is a short exact sequence describing
the relation between the controlling algebra of relative Rota-Baxter operators
on a Leibniz algebra with respect to a symmetric (resp. antisymmetric)
representation and the controlling algebra of the induced relative Rota-Baxter
operators (resp. averaging operators) on the canonical Lie algebra associated
to the Leibniz algebra. Finally, we show that there is a long exact sequence
describing the relation between the cohomology groups of a relative Rota-Baxter
operator on a Leibniz algebra with respect to a symmetric (resp. antisymmetric)
representation and the cohomology groups of the induced relative Rota-Baxter
operator (resp. averaging operator) on the canonical Lie algebra.
