Q. Let $C$ be a space such that $C=A \cup B$ and $A \cap B ↪ A$ is a cofibration. Show that $B ↪C$ is a cofibration.
Attempt : Let Y be any space. Let us denote the map $B ↪C$ by $i.$ Let $F:B \times I \to Y$ and $g:C \to Y$ be continuous maps such that $F(x,0)=(g \circ i)(x) \; \forall \; x \in B.$
Since $A \cap B ↪ A$ is a cofibration, denote it by $j$, observe that $i$ restricted to $A \cap B$ is equal to $j$. Similarly, $F$ restricted to $(A \cap B) \times I$ and $g$ restricted to $A$ are continuous maps such that $F(x,0)=(g \circ i)(x) \; \forall \; x \in A \cap B$.
Using the fact that $j$ is a cofibration, there exists a homotopy $H:A \times I \to Y$ such that $(H \circ i)=F$ and $H(x,0)=g(x) \; \forall \; x \in A$.
What I want to do is extend $H$ to a homotopy on $C \times I$ but I am clueless as to what information should be used to achieve this. I am looking for some hints to this end. Thank you.