The classical lefschetz fixpoint theorem is stated for oriented compact manifolds $M$ and a smooth map $f:M\to M$ as follows: the intersection number $I(\Delta, \mathrm{graph}(f))$ is equal to the lefschetz number $L(f)$.

First it is clear, that the intersection number at a fixpoint is independent of the orientation attached to a neighborhood of the fixpoint. So the intersection number makes sense.

There is the weaker lefschetz fixpoint theorem for finite simplicial complexes, which also applies in this case, where you only get the existence of fixpoints.

The question is if the above version with the intersection number holds for non-orientable manifolds too.