Mapping torus for $\sigma:\; S^1 \longrightarrow S^1$ as a pushout of $S^1 \subset D^2$ and some $\varphi:\; S^1 \rightarrow S^1 \vee S^1$


but how do I glue $D^2$ on this to obtain a mapping torus for $\sigma$? Can someone (help me) make $\phi$ explicit? I'm studying for an exam, this is not homework. The lecture notes simply don't say anything about how the pushout is constructed. Please be explicit and avoid 'words about images', if possible, I'd like to have formulas.
$D^2 \cong I^2$ might help here, I guess...

P.s.: This is not central to the topic right now, either. What I want to prove is just part of another proof concerning the homological degree...

The pushout diagrams would commute through usage of the homeomorphy \begin{align} \phi:&\;D^2 \longrightarrow S^1 \\ & x \mapsto \begin{cases} 0,\;x=0 \\ \frac{\|x\|_2 x}{\|x\|_1},\; x \ne 0 \end{cases} \end{align} which restricts to a homeomorphism between the boundaries.