Automorphism Group of a Space

Question

I am considering the following situation:

Let us call the space on the left $Y$ and the space on the right $X$.

The group of deck transformations is defined as $\operatorname{Aut}(Y/_X):=\{φ:Y→Y∣φ\text{ a homeomorphism s.t. }p∘φ=p\}$.

In this example, $\operatorname{Aut}(Y/_X):=\{\operatorname{Id}\}$. Could someone explain why?

Thank you in advance!