Understanding the defintion of dual operators

I'm reading a book about Functional Analysis, and now I've reached to the part about dual operators.

I'm having some difficulties understanding the following definition -

Why $$A^*$$ is $$Y^*\rightarrow X^*$$? We know that $$\phi \in Y^*$$, i.e., $$\phi:Y\rightarrow Y$$ and bounded. So the image of $$\phi(Ax)$$ should be in $$Y^*$$, shouldn't it?

How come it's in $$X^*$$?

($$X^*$$ is reffered here as the space of all bounded linear functionals on $$X$$).

First of all, $$\phi$$ is a functional, so it is $$\phi:Y\to\mathbb{C}$$.
Next, what is $$A^*$$? It takes a functional $$\phi\in Y^*$$ and returns a function $$A^*(\phi)=f$$ which is defined by $$f(x)=\phi(A(x))$$. If we take a vector $$x\in X$$ then $$A(x)\in Y$$ and then $$\phi(A(x))\in\mathbb{C}$$. So $$f:X\to\mathbb{C}$$. Then of course you need to check that $$f$$ is linear and bounded, but that's easy. So $$f\in X^*$$.