Modified traces and the Nakayama functor

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.