Let $(X,\le)$ be a complete ordered set with $f:X\to X$ monotone and suppose there are $a,b\in X$ such that $a\le f(a)\le f(b)\le b$.
Now, I can see there exists $c$ between $a $ and $b $ such that $c=f (c) $. But my book also states that $c$ is the infimum of $Y=\{x\in X | a\le x\le b, x\le f(x) \}$, which I can't get. We should have $c\le y $ for all $y\in Y$ but as far as I can tell, $a\in Y $ and $c\not\le a $.
I bet it's a typo, either in the book or in your quote from the book. This $c$ is the supremum of $$Y=\{x\in X\mid a\le x\le b,x\le f(x)\},$$ while also the infimum of $$Z=\{x\in X\mid a\le x\le b,x\ge f(x)\}.$$