Homology functors preserve coproducts I am trying to understand the proof that homology functors preserve coproducts (using Eilenberg-Steenrod Axioms) from here.
Here is the definition of admissible category for homology theory
Now, here is the proof that homology functors preserve finite coproducts
Here are my questions 


*

*The proof says that because of the 5-lemma we can just consider the case when $ A_1=A_2= \emptyset$. I don't see why this is true?

*In the proof it is assumed that the pairs $(X, X_1)$ and $(X,X_2)$ as well as the morphisms $f_1$ and $f_2$ are in the admissible category C. I don't see why this must be the case. We only know that  $(X_1, \emptyset)$,$(X_2, \emptyset)$ and $(X, \emptyset)$ belong to C. But why does that imply  $(X, X_1)$ and $(X,X_2)$ as well as the morphisms $f_1$ and $f_2$ are in the admissible category C? Also, it is not clear to me why the coproducts in C and $\bf{Top}_{(2)}$ are the same?
 A: Aboute the first part. The embeddings $(X_i,A_i) \to (X_1 \amalg X_2,A_1 \amalg A_2)$ induce the following diagram of long exact sequences.
$$\require{AMScd}
\begin{CD}
H_{n}(A_1) \oplus H_{n}(A_2) @>>> H_n(X_1) \oplus H_n(X_2) @>>> H_n(X_1,A_1)\oplus H_n(X_2,A_2) @>>> H_{n-1}(A_1)\oplus H_{n-1}(A_2) @>>> H_{n-1}(X_1)\oplus H_{n-1}(X_2)\\
@VVV @VVV @VVV @VVV @VVV \\
H_n(A_1 \amalg A_2) @>>> H_n(X_1 \amalg X_2) @>>> H_n(X_1 \amalg X_2,A_1 \amalg A_2) @>>> H_{n-1}(A_1 \amalg A_2) @>>> H_{n-1}(X_1 \amalg X_2)
\end{CD}
$$
So once you have proven the lemma for all the pairs of the form $(Y,\emptyset)$ you gain that the first, second, fourth and fifth vertical morphisms above are isomorphisms, and so by the five-lemma also the remaining morphism is an isomorphism too.
About the second part. I'm not sure about why $(X,X_1)$ and $(X,X_2)$ should belong to $\mathbf C$, notheless once you solve this problem by condition $ii)$ it follows that the embeddings $(X_j,\emptyset) \to (X,\emptyset) \to (X,X_j)$, and so their composites $f_j$'s, do belong to $\mathbf C$.
I'm aware that this doesn't solve all of your problems but I hope it could help.
A: On your second question, there is no reason at all that $(X,X_1)$ and $(X,X_2)$ should be in $\mathbf{C}$ just from the stated assumptions, or that coproducts in $\mathbf{C}$ coincide with coproducts in $\mathbf{Top}_{(2)}$.  In fact, it is easy to see that if $\mathbf{C}$ is any admissible category, so is the full subcategory $\mathbf{C}_0$ of $\mathbf{C}$ consisting of objects $(X,A)$ where either $A=\emptyset$ or $A=X$.  In particular, $(X,X_1)$ and $(X,X_2)$ will not be objects of $\mathbf{C}_0$ unless $X_1=\emptyset$ or $X_2=\emptyset$.
I think the hypotheses you want to assume in Theorem 2.3 is that $\mathbf{C}$ is closed under coproducts in $\mathbf{Top}_{(2)}$, and they still are coproducts (i.e., the inclusion maps into a coproduct in $\mathbf{Top}_{(2)}$ are in $\mathbf{C}$, and if you have two maps in $\mathbf{C}$ with the same codomain, the corresponding map in $\mathbf{Top}_{(2)}$ from their coproduct is also in $\mathbf{C}$).  Note that this guarantees that $(X,X_1)$ exists (and similarly for $(X,X_2)$), because $(X,X_1)$ can be described as the coproduct of $(X_2,\emptyset)$ and $(X_1,X_1)$.
As for the maps $f_1$ and $f_2$, while they can be shown to come from maps of $\mathbf{C}$, in the proof of Theorem 2.3 they are just maps in the category $\mathbf{A}$, defined by the formula $f_1=(j_2)_*(i_1)_*$ and $f_2=(j_1)_*(i_2)_*$.  Of course, this implies that they are induced by the composites $j_2i_1$ and $j_1i_2$.
