Equivalent versions of the Mayer-Vietoris axiom in Brown theorem

Question

In the hypotheses of Brown representability theorem there is a contravariant functor F from pointed connected CW complexes to pointed sets, which must respect two axioms, the second of which is the so-called “Mayer Vietoris” axiom. There are apparently three different versions of this axiom, which I presume are all equivalent, but why?

First version F sends homotopy pushouts to weak pullbacks

Second version $F(A \cup B) \to F(A)_{xF(A \cap B)} F(B)$ (where $F(A)_{xF(A \cap B)} F(B)$ is the homotopy fibre product of $F(A)$ and $F(B)$) is surjective.

Third version For any CW-triad $(X, A_1, A_2)$ and for any $x_1 \in F(A_1)$, $x_2 \in F(A_2)$, $(x_1)|_{A_1 \cap A_2} = (x_2)|_{A_1 \cap A_2}$, there is $y \in F(X)$ with $y|_{A_1} = x_1$ and $y|_{A_2} = x_2$

Answer 1

There's evidently a relation between the three conditions, considering that

1. a CW-triad is a triple of spaces $(X,A_1,A_2)$ such that $X = A_1\cup A_2$, and
2. a weak pullback is "a pullback with just existence, not uniqueness" -a condition that translates into a certain comparison map being surjective-.
3. You also need a small set of other pieces, like for example the relationship between cofibrations and homotopy pushouts (an homotopy pushout of a cospan, where one leg is a cofibration, is computed as a regular pushout). Now, the square $$ \begin{array}{ccc} A\cap B &\to & A \\ \downarrow && \downarrow \\ B &\to& A\cup B \end{array} $$ is a pushout (and a pullback, for that matter), and the inclusions of $A\cap B$ in $A,B$ are cofibrations (am I right? Just 99% sure), so the above remark applies.

In slightly more modern terms, the axioms for which Brown representability holds are these two:

(the categorical terminology in this pictureis proper of $\infty$-categories, but for the sake of the discussion here what matters is only that this allows you to remove the prefix "homotopy-" from all pushouts; plus, this is a particularly sleek statement of BRT; the part I removed is just a boilerplate assumption that the category of CW complexes satisfies, and that here one must impose on $\mathcal C$).

So, let's take $(b)$ above as the "standard" MV property; if $(b)$ is true, now, using my observation 3. you get condition 2 of your list. Now, a little bit of yoga (to be precise, first Virabhadrasana) unwinding the definition of pullback translates the second into the third condition of your list.