We express studies of the fuzzy relational compositions motivated
by their applications to medical diagnosis by Willis Bandler and
Ladislav Kohout. These types of compositions use only two
quantifiers: universal quantifier and existential quantifier. There
exists a huge gap between these quantifiers. This is a natural
motivation for studying fuzzy relational compositions defined based
on generalized quantifiers. In this paper, we revisit and provide
definitions of these compositions as well as those of derived
images and preimages. Furthermore, we investigate their
mathematical properties.