Gradual and fuzzy subsets

In fuzzy theory of sets and groups, the use of $\alpha$–levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong $\alpha$–levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. The main result of this paper is to identify the class of fuzzy sets, respectively fuzzy groups, with subcategories of the functorial categories $\mathcal{S}{\mathit{\text{et}}}^{(0,1]}$, resp. $\mathcal{G}{\mathit{\text{r}}}^{(0,1]}$.