  When trying to cast the free fermion in the framework of functorial field
theory, its chiral anomaly manifests in the fact that it assigns the
determinant of the Dirac operator to a top-dimensional closed spin manifold,
which is not a number as expected, but an element of a complex line. In
functorial field theory language, this means that the theory is twisted, which
gives rise to an anomaly theory. In this paper, we give a detailed construction
of this anomaly theory, as a functor that sends manifolds to
infinite-dimensional Clifford algebras and bordisms to bimodules.
