In this paper we deal with the problem of the relationship between the equipotency defined as a consequence of a given definition of scalar cardinality of a fuzzy set and the existence of a bijection between fuzzy sets, which also depends on the definition of fuzzy function or fuzzy mapping between fuzzy sets. We investigate the main definitions and we introduce an intermediate one that is a necessary and sufficient condition for the equipotency defined by every scalar cardinality in such a way that the partial results in[6], [8], [4] are include