Correspondence as a graph of a multifunction Suppose I'd like to say that a projection of $R\subset X\times Y$ on $X$ is the whole $X$. That is, $R$ is a graph of a certain multifunction, or equivalently it is a left-total relation. I do remember seeing somewhere the term correspondence being used exactly for such purposes. Is such terminology commonly used in set theory? Could you advise some classical text books where it is used?

Multivalued function is exactly the thing in the question. "Left-total relation" is not nearly as commonly used, although many readers might be able to guess that it means the same thing.  In some situations, multivalued function has more specific meanings such as a differential 1-form, algebraic function or branched covering, but these are compatible (or can be made so) with the set-theoretic idea of a multifunction.

term correspondence ... commonly used in set theory?

It has been standard in mathematics for a very long time, to use correspondence to mean a subset of $A \times B$, and the formal definition of inverses and composition of correspondences.  Subset can be generalized to a context-specific notion of sub-object (such as subvariety, submanifold, subspace, etc) of the product, maybe with additional properties such as being closed, and not necessarily the same thing as sub-objects in the category from which the $A$ and $B$'s are taken.
Common special cases are:


*

*one-to-one correspondence (bijections) 

*many-to-one correspondences (functions)

*one-to-many correspondences (multivalued functions) 

*correspondences in algebraic geometry

