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 $C_bullet(X,A cup B) overset{tilde{Delta}}{longmapsto} C_bullet(X,A) otimes C_bullet(X,B)$.

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 $C_bullet((X,A) times (Y,B)) = C_bullet(X times Y) / (C_bullet(A times Y)+ Cbullet(Xtimes B))$ 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.