  In this paper, we introduce the notion of the pivotal cover
$\mathcal{C}^{\mathsf{piv}}$ of a left rigid monoidal category $\mathcal{C}$ to
develop a theoretical foundation for the theory of Frobenius-Schur (FS)
indicators in "non-pivotal" settings. For an object $\mathbf{V} \in
\mathcal{C}^{\mathsf{piv}}$, the $(n, r)$-th FS indicator $\nu_{n,
r}(\mathbf{V})$ is defined by generalizing that of an object of a pivotal
monoidal category. This notion gives a categorical viewpoint to some recent
results on generalizations of FS indicators.
  Based on our framework, we also study the FS indicators of the "adjoint
object" in a finite tensor category, which can be considered as a
generalization of the adjoint representation of a Hopf algebra. The indicators
of this object closely relate to the space of endomorphisms of the iterated
tensor product functor.
