What's the meaning of $\tilde{y}$? In Linear Algebra  by Liesen, J. and Mehrmann, V., the following notation is used:

This chapter is about Maps in Set Theory. The authors explain the meaning of each symbol, nevertheless, they missed this one. I did some research in Wolfram Math World, but none of the examples of how the tilde is used throughout math seems promising for this example (at least none that I can make sense of).
Here is a screenshot of where it is first used in the book:


They said "Let $\tilde{y} \in Y$ be given". This means they're just picking an arbitrary element of the set $Y$. One can just as well say "Let $\xi \in Y$ be given", or "Let $\zeta \in Y$ be given" or "Let $\ddot{\smile} \in Y$ be given". It's just a symbol and a name which we temporarily give to the arbitrarily chosen element of the set $Y$.
For example, I can write the bottom of the proof as:

Let $\ddot{\smile} \in Y$ be given, then
  \begin{align}
(f \circ g)(\ddot{\smile}) &= f(g(\ddot{\smile})) = f(x_{\ddot{\smile}}) = \ddot{\smile}
\end{align}
  Hence $f \circ g = \text{Id}_Y$.

From a mathematical perspective, this is a perfectly meaningful proof, because from a purely mathematical perspective, what's to stop you from calling something a smiley-face? Aboslutely nothing at all! (of course, in practice no one actually writes a smiley face for elements of a set...) 
