Can anyone please help me with question?
Question: Let $p: E \rightarrow B$ be a fibration. Suppose there is a commuting diagram $$\begin{equation*}\begin{array}{ccl} (\mathbb{I}\times\{0\}\bigcup\{0,1\} \times \mathbb{I}) \times X&\stackrel{H}{\longrightarrow}&E\\ \!\!\!\!\!\!\!\!{\scriptstyle}\downarrow&&\downarrow{\scriptstyle p}\\ \!\!\!\!\mathbb{I} \times \mathbb{I} \times X&\stackrel{\phi}{\longrightarrow}&B \end{array}\qquad\qquad\end{equation*}$$
Show that $\phi$ has a lift $\phi^\star$ to $E$ extending $H$. Notation $\mathbb{I}= [0,1]$.
Thanks.
$f:\mathbb{I}\times \{0\}\times X\rightarrow \mathbb{I}\times \mathbb{I}\times X$ and $g:\mathbb{I}\times \{0,1\}\times X\rightarrow \mathbb{I}\times \mathbb{I}\times X$ are weak cofibrations, if the diagram commutes, the following diagrams commute
$$\begin{equation*}\begin{array}{ccl} (\{0,1\} \times \mathbb{I}) \times X&\stackrel{H}{\longrightarrow}&E\\ \!\!\!\!\!\!\!\!{\scriptstyle}\downarrow&&\downarrow{\scriptstyle p}\\ \!\!\!\!\mathbb{I} \times \mathbb{I} \times X&\stackrel{\phi}{\longrightarrow}&B \end{array}\qquad\qquad\end{equation*}$$
$$\begin{equation*}\begin{array}{ccl} (\mathbb{I}\times\{0\} ) \times X&\stackrel{H}{\longrightarrow}&E\\ \!\!\!\!\!\!\!\!{\scriptstyle}\downarrow&&\downarrow{\scriptstyle p}\\ \!\!\!\!\mathbb{I} \times \mathbb{I} \times X&\stackrel{\phi}{\longrightarrow}&B \end{array}\qquad\qquad\end{equation*}$$
Since in a Quillen model, the class of maps which has the left lifting property in respect of the class of fibrations is the class of weak cofibrations, you can find a lift $\phi_1$ and $\phi_2$ of the two previous diagrams which will enable you to construct $\phi^*$.