As the question suggests, given two random variables $X, Y$, I am wondering if there is a special name for the relation $E[X|Y]=Y$ (we know that $E[X|Y]=f(Y)$ for some measurable function $f$, so another way to ask the question is if there is a name for the special case when $f$ is the identity function).
Context: I am interested in the particular case where $X$ is an indicator function. In that case, $E[X|Y]=P(X|Y)=Y$, which we can understand roughly as saying $Y$ is a good estimator for the probability of $X$ absent further information.
Any help would be much appreciated.