Consider the following situation for a map $p : E → B$. We are given a homotopy $F : X × I → B$, and a map $g : X → E$ such that, $p ◦ g(x) = F(x, 0), ∀ x ∈ X$. We say $p$ satisfies the homotopy lifting property if for each homotopy lifting data as above, there exists a homotopy $G : X × I → E$ with the property $p ◦ G = F$ and $G(x, 0) = g(x), ∀ x ∈ X$
I have two questions:
$1.$ How can there be multiple homotopy liftng data's? What exactly represents the set of homotopy lifting data from which you can choose a particular homotopy lifting data? (Referencing the bolded text in the above box)
$2.$ Aren't these graphs incorrect since $g: X \rightarrow E$ and not $g: X \times \{0\} \rightarrow E$?