Im wondering if there's a short way to prove that when two manifolds with diffeomorphic boundaries are glued together along the boundaries the orientations of these must be inverse to each other. That is to say, suppose you have $M$ and $N$ oriented $n$-dimensional manifolds with $\partial M \cong \partial N$ under a diffeomorphism $\phi: \partial M \to \partial N$, you form $ C = M \cup_\phi N$, in order to do this you need that the orientation of $\partial M$ be opposite to that of $\partial N$, why is that? By homological means...

I understand the reason via the orientation of the tangent spaces and the outward-first orientation of the boundaries, but how can i prove it with fundamental classes? I know the homology of the pair $(C,\partial M) \cong (M,\partial M) \oplus (N,\partial N)$ (relative Mayer-Vietoris) and the inclusion $j: (C, \emptyset) \to (C,\partial M)$ induces a monomorphism in the top homology because of the exact sequence of the pair $(C,\partial M)$, and I came up with a "proof" using this, but it is way to lengthy, maybe there is a "quick way" to do this?

Thanks

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