I am currently reading Lemma 9.11 on Switzer, in the chapter dedicated to Brown representability theorem. However I am stuck on a few points of the proof.
Let $F$ a contravariant functor from pointed CW complexes up to homotopy to pointed sets. Let $Y$ be a pointed CW complex, $u \in F(Y)$ and $(X, A, x_0)$ a CW-pair. Let $g : (A, x_0) \to (Y, y_0)$ a cellular map. Let $T =( I^{+} \wedge A) \vee X \vee Y$, by identifying $[0,a] \in I^{+} \wedge A$ with $a \in X$ and $[1,a] \in I^{+} \wedge A$ with $g(a) \in Y$. Let $A_1 =( [0, 1/2]^{+} \wedge A) \cup X$ and $A_2 =( [1/2, 1]^{+} \wedge A) \cup Y$. Now $A_1 \cup A_2 = T, A_1 \cap A_2$ is isomorphic to $A$. Until this point it is all clear. What I don’t understand is the following:
There is a strong deformation retraction $f: A_1 \to X$ and $Y$ is a strong deformation retract of $A_2$. Thus there are $\bar{v} \in F(A_1)$ with $\bar{v}|_{X} = v$ and a $\bar{u} \in F(A_2)$ with $\bar{u}|_{Y}= u$. Why is that?
Your space $T$ is nothing else than the reduced double mapping cylinder of the maps $\iota : A \hookrightarrow X$ and $g : A \to Y$. To see that, note that $I^+ \wedge A = (I \times A)/(I \times \{a_0\})$.
The subspaces $A_1$ and $A_2$ of $T$ are copies of the reduced mapping cylinders of $\iota$ and of $g$. Thus you get the usual strong deformation retractions $f : A_1 \to X$ and $r : A_2 \to Y$. Hence the inclusions $i_1 : X \to A_1$ and $i_2 : Y \to A_2$ are homotopy equivalences and induce bijections $i_1^* : F(A_1) \to F(X)$ and $i_2^* : F(A_2) \to F(Y)$.
Now consider $v \in F(X)$ and $u \in F(Y)$.
Let $\bar v = (i_1^*)^{-1}(v) \in F(A_1)$. Then by definition $\bar v \mid_X = i^*_1(\bar v) = i^*_1( (i_1^*)^{-1}(v)) = v$.
Let $\bar u = (i_2^*)^{-1}(u) \in F(A_2)$. Then $\bar u \mid_Y = u$.