  We demonstrate that the concept of strict proto-differentiability of
subgradient mappings can play a similar role as smoothness of the gradient
mapping of a function in the study of subgradient mappings of prox-regular
functions. We then show that metric regularity and strong metric regularity are
equivalent for a class of generalized equations when this condition is
satisfied. For a class of composite functions, called C2-decomposable, we argue
that strict proto-differentiability can be characterized via a simple relative
interior condition. Leveraging this observation, we present a characterization
of the continuous differentiability of the proximal mapping for this class of
function via a certain relative interior condition. Applications to the study
of strong metric regularity of the KKT system of a class of composite
optimization problems are also provided.
