In this paper, we first consider the disjoint union of two fuzzy metric spaces and non-existence of fuzzy contractive maps. We prove that there exists a natural fuzzy metric on the disjoint union of two fuzzy metric spaces. We also show that there is a fuzzy metric space which does not admit any fuzzy contractive map. Second, we introduce the notion of a fuzzy diameter for fuzzy metric spaces and obtain some properties on it and its relations to classical diameter for compact metric spaces. Moreover, using the notion of fuzzy diameter, we construct a fuzzy metric on the disjoint union of two fuzzy metric spaces. Finally, we consider the fuzzy product metric space of two fuzzy metric spaces and prove that there are fuzzy metrics on the product space such that the inclusions into the product space become fuzzy isometric embeddings. This shows that there is a family of fuzzy isometric embeddings from a fuzzy metric space into another fuzzy metric space.