Hello, I'm having problems undestanding an example. It's supposed to be simple, but I can't quite draw the conclusions.

Let there be a Set, $\displaystyle M$ and let there be $\displaystyle Map(M, M) := \{f~|~ f: M \rightarrow M$ is a mapping$\displaystyle \}$. If $\displaystyle f, g \in Map(M, M)$, so is $\displaystyle f \circ g$ also in $\displaystyle Map(M, M)$. And so is $\displaystyle \circ $ a composition of $\displaystyle Map(M, M)$.

My question is, what exactly is $\displaystyle Map(M, M)$? Is it the set $\displaystyle M$x$\displaystyle M$? And are $\displaystyle f$ and $\displaystyle g$ therefore $\displaystyle Map(f, f')$ and $\displaystyle Map(g, g')$? Because, in the set definition $\displaystyle Map(M, M) := \{f~|~ f: M \rightarrow M\}$, you have a mapping from $\displaystyle M \rightarrow M$?