For the simple purpose of taking notes without writing too much, I'm wandering two things.
Given the category $C$
- is it ok to write $c \in C$ to express the idea that the object $c$ is in the category $C$?
- Is there a symbol to express that $f: a \to b$ is a morphisms between objects of $C$? I guess even if the answer to the previous question was yes, $f \in C$ would be ambiguous because the reader wouldn't know whether $f$ is a morphism or an object.
As far as I know, the symbol $\in$ is a thing in set theory at least, but I'm not sure if it's appropriate for category theory, hence I'm asking this question.
A common notation seems to be $$c\in\operatorname{Ob}(C)$$ and $$f\in\operatorname{Mor}(C)$$ (or if you want to be more specific: $f\in \operatorname{Mor}_C(a,b)$).
Using $\in$ in this context is okay because $\operatorname{Ob}(C)$ and $\operatorname{Mor}(C)$ are at least classes, though typically not sets.