Fuzzy functions based on adjoint relations

We apply the theory of adjoint functors in category theory to extend the properties of classical functions to the case of fuzzy functions. Firstly, the composition properties of fuzzy functions defined by fuzzy equalities are discussed. Secondly, by employing the adjoint relation between a fuzzy function and its inverse function, the membership function of the preimage of fuzzy functions defined by Demirci is redefined. Finally, based on the redefined membership function, the related properties of images, preimages, and the preservation of operations under fuzzy functions are investigated.