Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator $P\colon V\to C(X)$ from a Hausdorff separable locally convex space $V$ to the Banach space $C(X)$ of continuous functions on a compact set $X$ we prove that the subdifferential $\partial P$ at zero is operator-affinely homeomorphic to the compact subdifferential $\partial^cQ$, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator $Q\colon\ell^2\to C(X)$ from a separable Hilbert space $\ell^2$, where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space $L^c(\ell^2,C(X))$ of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.