I cannot find any categorical definition of an eigenvalue, so I ask this question. Let $\mathbb{k}$ a be a field and $\mathcal{C}$ be a $\mathbb{k}$-linear abelian category. Let $f: X \rightarrow X \in \mathrm{End}_\mathcal{C}(X)$. To me, it makes sense to call $\lambda \in \mathbb{k}$ an eigenvalue of $f$ if $\ker(f - \lambda 1_X)$ is nonzero (and call this the corresponding eigenspace). By considering pullbacks one can show that these kernels do not "intersect" either for different $\lambda$.

If this indeed is the accepted definition, what are some reasonable set of conditions so that any such $f$ always has an eigenvalue (for instance, algebraic closedness of $\mathbb{k}$ will probably be necessary and some finiteness assumption)?

The greater context for such a question is from trying to prove categorical Schur's lemma for a tensor category, where any endomorphism of a simple object is a scalar multiple of the identity. And a similar statement about an endomorphism of an indecomposable being of the form $\lambda 1_X + n$, where $n$ is nilpotent.