This is the continuation of Montes' paper "On the canonical discussion of polynomial systems with parameters''. In this paper, we define the Minimal Canonical Comprehensive Gröbner System of a parametric ideal and fix under which hypothesis it exists and is computable. An algorithm to obtain a canonical description of the segments of the Minimal Canonical CGS is given, thus completing the whole MCCGS algorithm (implemented in Maple and Singular). We show its high utility for applications, such as automatic theorem proving and discovering, and compare it with other existing methods. A way to detect a counterexample to deny its existence is outlined, although the high number of tests done give evidence of the existence of the Minimal Canonical CGS.