I'm working through a proof that, given an oriented compact (connected) $n$-manifold $M$ with boundary, any two continuous maps $f,g:M\to S^n$ are homotopic. The proof uses the double of $M$, which is obtained by gluing together two copies of $M$ at the boundary to obtain a smooth, compact, closed, oriented $n$-manifold; call it $M'$. Let $\pi:M'\to M$ be the natural projection and $\iota: M\to M'$ the natural embedding. Then, $deg (f\circ \pi)=0$ and so $f=f\circ \pi\circ \iota$ is homotopic to a constant map.
Why is $deg(f\circ \pi)=0$?