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?

It is standard terminology in mathematical economics. See for example, Aliprantis & Border 2007 Infinite Dimensional Analysis: A Hitchhiker's Guide. Terminology varies as to whether any subset of $X\times Y$ is a correspondence or whether the projectiont to $X$ has to be surjective (a "nonempty-valued correspondence"). Also, some people define it as a relation and some as a function with sets as values. Mathemtical economics and related fields such as optimization theory are some of the main users of the concept.
The term originated probably with Nicolas Bourbaki and was imported into economics by Gerard Debreu. In Bourbaki's book on sets, correspondences are treated in Chapter II §3, where a correspondence is defined as a triple of sets $(G,A,B)$ with $G\subseteq A\times B$.
