\(\operatorname {Hom}\) in \(\operatorname {Hom}_{{\cal C}}(X, Y)\) is short for homomorphism, since an arrow in category theory is a morphism (i.e. an arrow), a generalization of homomorphism between algebraic structures.
This notation could be unnecessarily verbose, so when there is no confusion, we follow [leinster2016basic] and [zhang2021type] to simply write \(X,Y \in \operatorname {Ob}({\cal C})\) as
\[X,Y \in {\cal C}\]
and \(f \in \operatorname {Hom}_{{\cal C}}(X, Y)\) as
\[f \in {\cal C}(X, Y)\]
In some other scenarios, when the category in question is clear (and it might be to too verbose to write out, e.g. a functor category), we omit the subscript of the category and write just
\[\operatorname {Hom}(X, Y)\]
In general, collection \(\operatorname {Ob}({\cal C})\) and \(\operatorname {Arr}({\cal C})\) are not neccessarily sets, but classes. In that case, \(\operatorname {Hom}_{{\cal C}}(X, Y)\) is called a hom-class.
Later, we will also learn that \(\operatorname {Ob}\) and \(\operatorname {Arr}\) are representable functors.
and the identity is given by
