Let $p: \mathbb{R}^2 \rightarrow S^1 \times S^1$ be the standard universal covering map of the torus. Let $x_o=p(0,0).$ Suppose $q \in \mathbb{R}P^2$. Show that composition with $p$ induces a one-to-one correspondence between continuous maps $(\mathbb{R}P^2,q) \rightarrow (S^1 \times S^1, x_0)$ and continuous maps $(\mathbb{R}P^2,q) \rightarrow (\mathbb{R}^2,(0,0))$
My proof feels like it is missing something... I have shown that given a map from one of these sets, it has a canonical way to persribe it a map in the other set. I'm not sure if I have completely and rigourously answered the question, can somebody help me out? Thanks!
$Proof:$
Every map $f: (\mathbb{R}P^2,q) \rightarrow (S^1 \times S^1, x_0)$ has a unique lift $\tilde{f}: (\mathbb{R}P^2,q) \rightarrow (\mathbb{R}^2,(0,0))$ such that $p\tilde{f}=f$.
Also, starting with a map $g: (\mathbb{R}P^2,q) \rightarrow (\mathbb{R}^2,(0,0))$, we have that $pg$ is a map from $(\mathbb{R}P^2,q) \rightarrow (S^1 \times S^1, x_0)$.