<p>In this paper, we consider a class of nonlinear equations derived from first-order type methods for solving composite optimization problems. Traditional approaches to establishing superlinear convergence rates of semismooth Newton-type methods for solving nonlinear equations usually postulate either nonsingularity of the B-Jacobian or smoothness of the equation. We investigate the feasibility of both conditions. For the nonsingularity condition, we present equivalent characterizations in broad generality and illustrate that they are easy-to-check criteria for some examples. For the smoothness condition, we show that it holds locally for a class of residual mappings derived from composite optimization problems. Furthermore, we investigate a relaxed version of the smoothness condition - smoothness restricted to certain active manifolds. We present a conceptual algorithm utilizing such structures and prove that it has a superlinear convergence rate.</p>