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Is there a wide-spread notation for identity morphism on a specified object of a specific category C? (I consider several categories and there are several different identity morphisms (one for each category) on the same object.)
$1_X$ or $id_X$ are frequently used.
I suggest creating some notation that distinguishes when you want to consider the object as living in one category, and as living in another. For instance, the set $S$ and the discrete topological space $S^\delta$ (not a great example, since the identity function is the same in both cases). If you don’t have a functor from one category to the other, then I would in fact strongly suggest notation that makes the objects obviously different. Abusing notation and omitting inclusion functors can sometimes be ok if people know what you’re talking about, but otherwise be as clear as you possibly can. (That’s all I’ve got)
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