I'm trying to get to grips with triangulated categories at the moment. According to Wikipedia, the shift/translation functor of a triangulated category $\mathcal{C}$ is an "additive automorphism (or for some authors, an auto-equivalence)" $\Sigma:\mathcal{C}\to\mathcal{C}$. It gives a link to the Wikipedia page Equivalence of Categories, which defines an auto-equivalence on a category $\mathcal{C}$ to be a functor $F:\mathcal{C}\to\mathcal{C}$ for which there exists a functor $G:\mathcal{C}\to\mathcal{C}$ such that $G\circ F$ and $F\circ G$ are 'naturally isomorphic' to the identity functor on $\mathcal{C}$.
In $\S 1.3$ of Triangulated Categories (Holm & Jørgensen, 2010), the shift/translation functor on $\mathcal{C}$ is described as "an additive functor which is an automorphism (i.e. it is invertible, thus there exists a functor $\Sigma^{-1}$ such that $\Sigma\circ\Sigma^{-1}$ and $\Sigma^{-1}\circ\Sigma$ are the identity functors)".
Unless I've missed something, one is far stricter than the other. In the latter case, $\Sigma$ must be invertible, and behave much like an isomorphism normally does. In the former case, there is a lot more flexibility, and being 'naturally isomorphic' to the identity functor will do.
Many other sources simply refer to an 'auto-equivalence' without stating which of these they refer to. I am fairly new to category theory so please forgive my naivety.