quick question for my better understanding:
Assume you have an additive category $\mathcal{C}$ and an automorphism $\Sigma$ of this category. Does $\Sigma$ send objects to isomorphic objects? If it doesn't in general, when can I expect it (under which reasonable assumptions) to do so? I'm asking this to clarify my understanding while reading about triangulated categories.
No, this is very rarely true even with extremely nice hypotheses on $C$. Consider the category of $\mathbb{Z}$-graded vector spaces and the automorphism $\Sigma$ which shifts the grading up by $1$. The objects $c$ such that $c \cong \Sigma c$ are the $\mathbb{Z}$-graded vector spaces each of whose graded pieces have the same dimension.
What's true is that $\Sigma c$ has the same categorical properties as $c$ (which is to say, properties that only depend on how $c$ sits inside $C$). For example, $c$ is projective iff $\Sigma c$ is projective. This is similar to how elements of, say, a ring need not be fixed by automorphisms of the ring, but two elements related by an automorphism share ring-theoretic properties such as being nilpotent.