I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can be what he states it to be. Let $\beta':S^1 \rightarrow I \wedge S^1$ be defined by $(2t,\ast)$ if $t \leq 1/2$ and $(\ast,2t-1)$ if $t > 1/2$. Consider the quotient map $q:I \rightarrow S^1$ given by $q(t) = e^{2\pi t}$. Switzer then claims that the composition $\alpha= (q \wedge 1) \circ \beta': S^1 \rightarrow S^1 \wedge S^1$ is given by $\alpha(t) = (4t,\ast)$ it $t \leq 1/4$, $\alpha(t) = (\ast,2t-1/2)$ if $1/4 \leq t \leq 3/4$ and $\alpha(t) = (4(1-t),\ast)$ if $3/4 \leq t \leq 1$. However, I get that the composition is $(2s,\ast)$ for $t \leq 1/2$ and $(\ast, 2t-1)$ for $t \geq 1/2$. Is Switzer wrong, or am I misunderstanding something? Any help would be very appreciated, I don't seem to be able to fix this detail.

I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can be what he states it to be. Let $\beta':S^1 \rightarrow I \vee S^1$ be defined by $(2t,\ast)$ if $t \leq 1/2$ and $(\ast,2t-1)$ if $t > 1/2$. Consider the quotient map $q:I \rightarrow S^1$ given by $q(t) = e^{2\pi t}$. Switzer then claims that the composition $\alpha= (q \vee 1) \circ \beta': S^1 \rightarrow S^1 \vee S^1$ is given by $\alpha(t) = (4t,\ast)$ it $t \leq 1/4$, $\alpha(t) = (\ast,2t-1/2)$ if $1/4 \leq t \leq 3/4$ and $\alpha(t) = (4(1-t),\ast)$ if $3/4 \leq t \leq 1$. However, I get that the composition is $(2s,\ast)$ for $t \leq 1/2$ and $(\ast, 2t-1)$ for $t \geq 1/2$. Is Switzer wrong, or am I misunderstanding something? Any help would be very appreciated, I don't seem to be able to fix this detail.