I am studying the book Algebraic Topology by Edwin Spanier. I do not understand how to prove Theorem 7.1.15, whose proof is left as an exercise to the reader, and would like some help. If it is important, all of the following spaces have a basepoint, maps are basepoint preserving, homotopies are basepoint preserving, etc; all constructions like the mapping cone and the suspension are modified in the standard way for homotopy theory of pointed spaces.
The question is as follows. Let $f : X'\to X$ be a map of spaces, and let $C_f$ denote the mapping cone. Let $\beta_1, \beta_2, \beta_3$ be all maps from $C_f$ into some space $Y$, subject to the constraint that $\beta_1 = \beta_2=\beta_3$ when restricted to $X\subset C_f$.
Define $Z$ to be the coproduct of $C_f$ with itself, glued together along their common copy of $X$. So if $i$ is the inclusion $X\subset C_f$, then $Z$ is the pushout of $i$ and $i$. Then $\beta_1$, $\beta_2$ agree on $X$, so they define a map $\beta_1 + \beta_2 : Z\to Y$. Likewise with $\beta_2,\beta_3$.
There is a natural map $j$ from the (reduced) suspension $S(X')$ to $Z$. If we view $S(X')$ as the union of two cones $C(X')$ glued along the common subspace $X'$, then one takes two copies the obvious map $C(X') \to C_f$ and look at the induced map on the pushouts. Formally let us define the suspension $S(X')$ as $X'\times I$ where $X'\times \left\{0\right\}$, $X'\times \left\{1\right\}$, and the basepoint $x_0\times I$ are all identified down to a single common point. We send the "top" endpoint of $S(X')$ to the top endpoint of $Z$; the bottom endpoint of $S(X')$ to the bottom endpoint of $Z$; and at $t=0.5$ we define the map $X'\times \left\{0.5\right\}\to Z$ to be $f : X'\to X\subset Z$.
I define $d(\beta_1,\beta_2)$ to be the map $(\beta_1+\beta_2)\circ j$. So $d(\beta_1,\beta_2): S(X')\to Y$. I define $d(\beta_2,\beta_3)$, $d(\beta_1,\beta_3)$ the same way.
Recall that there is a map $\kappa$ from $S(X') \to S(X')\vee S(X')$ for any space $X'$; the idea is to view $S(X')\vee S(X')$ as homeomorphic to $S(X')$ with the subspace $X'\times \left\{0.5\right\}$ identified down to a single basepoint where the two are joined. This quotient map is the map. This map makes $S(X')$ into a "cogroup" in the homotopy category; for any two homotopy classes of maps $[f], [g] : S(X')\to Y$, we can define a product $[f]\ast[g] := (f+g)\circ \kappa$. This endows $[S(X');Y]$ with a group structure.
Spanier claims that for any maps $\beta_1, \beta_2,\beta_3$ as above, $d(\beta_1,\beta_3) \simeq d(\beta_1,\beta_2)\ast d(\beta_2,\beta_3)$.
He does not prove this but leaves it as an easy exercise for the reader. I cannot draw myself a picture that convinces me that this homotopy is valid; and without the picture I am really not sure where to start with the algebra.