Categories with an endomorphism functor

Stream: maths

[Leucocholia]

There are some categories for which the endomorphism function End on objects forms a functor, and there's some research on this and I found some sufficient conditions, but are there any known necessary and sufficient conditions?
Building on that, what kinds of categories have a natural transformation exp : id => End and/or a natural transformation log : End => id (are they always unique and inverses)? How would you formalize this in Lean?