I am trying to define the degree of maps between torus. While proving some properties of degree, I came across something where I got stuck.
I have two maps $\psi, \phi:S^1\times S^1\to S^1$ with the property that $\phi\circ i_k$ is homotopic to $\psi\circ i_k$ where $i_k, k=1,2$ are the inclusion map taking $z\mapsto (z,1)$ and similarly $z\mapsto (1,z).$ I want to claim that this would imply that $\phi, \psi$ are homotopic. Is it true? And any help please. I am trying this for last two days, and have made no progress at all.