  We establish the deformation theory of Lie groupoid morphisms, describe the
corresponding deformation cohomology of morphisms, and show the properties of
the cohomology. We prove its invariance under isomorphisms of morphisms.
Additionally, we establish stability properties of the morphisms using
Moser-type arguments. Furthermore, we demonstrate the Morita invariance of the
deformation cohomology and consider simultaneous deformations of the morphism,
its domain, and codomain. These simultaneous deformations are utilized to
define cohomology for generalized morphisms and to study deformations of
multiplicative forms on Lie groupoids.
