Ah, I think I now have the same problem I had last time. The pairs are $(X,x_0)$ and $(Y,y_0)$. I send $([\varphi], [\psi])$ to $(i_X^\ast p_X^\ast([\varphi]) + i_X^\ast p_Y^\ast([\psi])$ and want to show the last thing is zero. $i_X^\ast p_Y^\ast = (p_Y i_X)^\ast = (X \rightarrow \{y_0\})^\ast$.
I think I understand something wrong. I try to calculate using a representant $\psi$, on cochain level. Then for an arbitrary sum of simplices $\sum_i a_i \sigma_i $, I get $ i_X^\ast p_Y^\ast \psi(\sum_i a_i \sigma_i) = \sum_i a_i\psi(\Delta^k \rightarrow y_0)$ and duh is there an immediate reaso…