For a category $\mathbb{C},$ with identity functor $\mathbb{C} \overset{1_\mathbb{C}}{\rightarrow} \mathbb{C},$ I have recently found that natural transformations $1_\mathbb{C} \overset{\alpha}{\rightarrow} 1_\mathbb{C}$ can be very interesting. Mainly I am wondering if such natural transformations have been studied before. Do they have a name ? Can anybody recommend literature or important results about them to look at?
More concretely, I am also wondering what kinds of categories $\mathbb{C}$ have natural isomorphisms $1_\mathbb{C} \overset{\alpha}{\rightarrow} 1_\mathbb{C}$ other than the identity of $1_\mathbb{C}.$