I'm trying to show that a punctured torus deformation retracts to a wedge of 2 circles. So I considered a punctured solid square (which eventually becomes a torus after identification of opposites sides), and make it deformation retract onto its border (the wedge of circles by the same identification). So the argument I'd like to use is the fact that a deformation retraction of a given space $X$ onto a subspace $A$ induces a deformation retraction of $X/R$ onto $A/R$, whatever equivalence relation R is considered. Is that true in general? If so could anybody explain why?
I am tempted to induce in the following way: given $H:X*I\rightarrow X$, define $G:X/R*I\rightarrow X/R$ by $G([x],t)=[H(x,t)]$, but I can't show that this is well defined...