I'd like to understand relative cap product based on my notes. We want a map $$H^i(X,A) \otimes H_n(X,A \cup B) \longmapsto H_{n-i}(X,B)$$
In order to understand the construction, we want to give a map $C_\bullet(X,A \cup B) \overset{\tilde{\Delta}}{\longmapsto} C_\bullet(X,A) \otimes C_\bullet(X,B)$. But what's this map?
In fact, we can consider $0 \longmapsto C_\bullet(A) \longmapsto C_\bullet(X) \longmapsto C_\bullet(X,A) \longmapsto 0$ and tensor with $C_\bullet(X,B)$ to get
$$0 \longmapsto C_\bullet(A) \otimes C_\bullet(X,B) \longmapsto C_\bullet(X)\otimes C(X,B) \longmapsto C_\bullet(X,A) \otimes C_\bullet(X,B) \longmapsto 0$$
Now consider the following commutative diagram
$$\begin{array}{ccccccccc} C_\bullet(A) & \to & C_\bullet(X,B) & \to & C_\bullet(X,A \cup B) & \to & 0 \\\ \downarrow{\Delta} & & \downarrow{\Delta} & & \downarrow{\Delta} \\\ C_\bullet(A \times (X,B)) & & C_\bullet(A \times(X,B)) & & C_\bullet((X,A)\times (X,B)) \\\ \downarrow{EZ} & & \downarrow{EZ} & & \downarrow{EZ} \\\ 0 \to C_\bullet(A) \otimes C_\bullet(X,B) &\to & C_\bullet(X)\otimes C(X,B) & \to & C_\bullet(X,A) \otimes C_\bullet(X,B) & \to 0 \end{array}$$
I have the following questions:
$0$. Why having the map $C_\bullet(X,A \cup B) \overset{\tilde{\Delta}}{\longmapsto} C_\bullet(X,A) \otimes C_\bullet(X,B)$ we end up with the cap product $H^i(X,A) \otimes H_n(X,A \cup B) \longmapsto H_{n-i}(X,B)$ ?
$1$. The sequence on the first line how is it build? I think the map $C_\bullet(X,B) \longmapsto C_\bullet(X,A \cup B)$ is surjective, is this enough in order to get the sequence?
$2$. Why the diagram is commutative and how this could help to prove $0$?
$3$. The $EZ$ in the diagram, seem the relative Eilenberg Zielber maps (reference here).
If so, are $C_\bullet((X,A) \times (Y,B)) = C_\bullet(X \times Y) / (C_\bullet(A \times Y)+ C\bullet(X\times B))$ and $C_\bullet(A \times (X,B))$ defined as the previous with $A = (A,\varnothing)$?
$4$. Are the vertical $\Delta$s in reality the diagonal map composed with a projection?
Any help or reference would be appreciated.