The goal of this paper is to introduce and study the induced proximity on a fuzzy space due to the existence of a fuzzy proximity on another fuzzy space. Firstly, for every two complete lattices L, M, it is defined and studied the extension of the -proximity on the fuzzy space  to an -proximity on the fuzzy space   and the restriction of the -proximity on  to an -proximity on  Then it is obtained the relations between their closure operators in each case (for general case and for special case when  Secondly, it is reformulated the definition of the fuzzy function on where  is the lattice of the power set of a nonempty set  Moreover, if  denotes the family of  all complete lattices defined on    is the map  then it is shown that every -basic proximity  on    induces -basic proximity  on  and the map  translates the family of ‑closed fuzzy subsets in  into the family of the -closed fuzzy subsets in  Thirdly, it is shown that the family of the categories of -fuzzy basic proximity spaces on    is embedded in the category of -fuzzy basic proximity spaces on

the extension of the -proximity, the restriction of the -proximity, -fuzzy subsets, fuzzy function, -fuzzy topology, -fuzzy proximity.