A categorical approach to probability allows to put basic notions of probability into a broader mathematical perspective, to evaluate their roles, and mutual relationships. Classical probability theory and fuzzy probability theory lead to two particular categories and their relationship (in categorical terms) enable us to understand and explicitly formulate the difference between them. Using our previous results, we show that the category ID of D-posets of fuzzy sets provides a framework in which the transition from classical to fuzzy probability theory is the consequence of some natural assumptions imposed on classical notions. Probability domains are constructed via suitable cogenerators and we study the transition in terms of the fuzzification of classical Boolean cogenerator. We introduce two categories CP and FP of probability spaces and observables corresponding to the classical probability theory and the fuzzy probability theory, respectively. We show that CP is isomorphic to a subcategory of FP.