Limiting coderivative ($D^{\ast }S(\bar{u}\mid \bar{v})$)

The limiting coderivative is $D^{\ast }S(\bar{u}\mid \bar{v})(v^{\ast })=\{u^{\ast }:(u^{\ast },-v^{\ast })\in N_{\mathrm{gph}S}(\bar{u},\bar{v})\}$. The negative sign on the output dual is essential. For a smooth single-valued function it gives the transpose Jacobian acting on $v^{\ast }$.