While proving that orientable cover $M$ of a manifold non-orientable manifold $N$ factored through the orientation double cover, I got stuck in this following problem...
If $p:N→M$ is a covering, N orientable, then $ \pi_M∘\pi_1(p)=\pi_N $ where $\pi_N$ is the morphism $\pi_1(N)→Z/2$, sending loops to the orientation of the base point according to lifts in the orientation double cover. $\pi_M$ is also defined in similar fashion. The covering map $p$ induced $\pi_1(p):\pi_1(N)\to \pi_1(M) $.
Intuitively it is clear to me, but I want a precise proof, which I am missing somehow.