  Rota-Baxter operators and more generally $\mathcal{O}$-operators play a
crucial role in broad areas of mathematics and physics, such as integrable
systems, the Yang-Baxter equation and pre-Lie algebras. The main objects of
study in the paper are certain $\mathcal{O}$-operator morphisms on associative
algebras. The cohomology theory of an associative $\mathcal{O}$-operator
morphism is established. In development, we give the Cohomology Comparison
Theorem of an $\mathcal{O}$-operator morphism, that is, the cohomology of an
$\mathcal{O}$-operator morphism is isomorphic to the cohomology of an auxiliary
$\mathcal{O}$-operator. As applications, we also study the Cohomology
Comparison Theorem of a Rota-Baxter operator morphism (of weight zero) and an
associative $r$-matrice weak morphism as a particular case of
$\mathcal{O}$-operator morphisms.
