Homology of the torus

Question

I am coming from this post and i am fine with everything that happens there. I just want a few more details on the calculation of the effect of $d_2$. Following the way i learnt it we have to calculate the degree of the map $S^1\overset{\phi_{\delta \alpha}}{\rightarrow} S^1\vee S^1 \overset{c_{\alpha'}}{\rightarrow} S^1$ where $\phi_{\delta \alpha}$ is the attaching map given by $aba^{-1}b^{-1}$ and $c_{\alpha'}$ collapses the respective circles (so $c_{\alpha'} $ varies. Now im stuck at making explicit what this actually means. Intuitively it is quite clear what is going on but its hard for me to make it rigorous. Any help is apprechiated!