We present mathematical detail pertaining to the theory of soft-margin support vector classifiers, designated C-SVC, as used in scikit-learn. We discuss the character of C-SVC, particularly with regard to the penalty term. We construct the primal problem and, thereafter, derive the dual problem. We introduce the notion of nonlinear classifiers and describe the so-called kernel trick. Additionally, we show how the primal problem can be derived from the dual problem. The paper is the second in a series and is intended to be educational in nature.