The global Lefschetz number of $f$ vanishes. - Is this conterexample work?

The global Lefschetz number of $f$ vanishes. - Is this conterexample work?

I am hoping someone will be willing to help me take a look at if this
conterexample works?
Let $X$ be an oriented compact manifold and $f : X \to X$ a map. Suppose
$W$ is a compact oriented manifold with boundary $\partial W = X$ and $F :
W \to W$ a map whose restriction to the boundary is $f$. Then the global
Lefschetz number of $f$ vanishes.
Counterexample:
Consider $W = D$, the unit disk. And $X = \partial W = S^1$. $$f =
\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta
\end{pmatrix}.$$
Then the only fixed point is the origin, and clearly $df_x - I = f(x) - I$
would not vanish $\forall \theta \neq 2k\pi, k \in \mathbb{Z}$.