  We organize the modified trace theory with the use of the Nakayama functor of
finite abelian categories. For a linear right exact functor $\Sigma$ on a
finite abelian category $\mathcal{M}$, we introduce the notion of a
$\Sigma$-twisted trace on the class $\mathrm{Proj}(\mathcal{M})$ of projective
objects of $\mathcal{M}$. In our framework, there is a one-to-one
correspondence between the set of $\Sigma$-twisted traces on
$\mathrm{Proj}(\mathcal{M})$ and the set of natural transformations from
$\Sigma$ to the Nakayama functor of $\mathcal{M}$. Non-degeneracy and
compatibility with the module structure (when $\mathcal{M}$ is a module
category over a finite tensor category) of a $\Sigma$-twisted trace can be
written down in terms of the corresponding natural transformation. As an
application of this principal, we give existence and uniqueness criteria for
modified traces. In particular, a unimodular pivotal finite tensor category
admits a non-zero two-sided modified trace if and only if it is spherical.
Also, a ribbon finite tensor category admits such a trace if and only if it is
unimodular.
