The objects of \\(\textbf{Cat}\\) are small categories and the arrows are functors between them.

The objects of \\(\textbf{Set}\\) are sets and the arrows are maps between them.

\\(\textrm{Ob} : \textbf{Cat} \rightarrow \textbf{Set}\\) sends:

— small categories \\(C\\) to their underlying object sets \\(\textrm{Ob}(C)\\)

— functors \\(F : C \rightarrow C'\\) to their underlying object maps \\(\textrm{Ob}(F) : \textrm{Ob}(C) \rightarrow \textrm{Ob}(C')\\)

\\(\textrm{Ob}\\) is a functor because it sends identity functors to identity maps and composition of functors to composition of maps.

It is basically a "forgetful" functor because it "forgets" the arrows in small categories, and "forgets" the what functors between them do to those arrows. It also forgets the composition of arrows _within_ a small category, but preserves the composition of functors _between_ small categories (ie composition of arrows of \\(\textbf{Cat}\\)).

The objects of \\(\textbf{Set}\\) are sets and the arrows are maps between them.

\\(\textrm{Ob} : \textbf{Cat} \rightarrow \textbf{Set}\\) sends:

— small categories \\(C\\) to their underlying object sets \\(\textrm{Ob}(C)\\)

— functors \\(F : C \rightarrow C'\\) to their underlying object maps \\(\textrm{Ob}(F) : \textrm{Ob}(C) \rightarrow \textrm{Ob}(C')\\)

\\(\textrm{Ob}\\) is a functor because it sends identity functors to identity maps and composition of functors to composition of maps.

It is basically a "forgetful" functor because it "forgets" the arrows in small categories, and "forgets" the what functors between them do to those arrows. It also forgets the composition of arrows _within_ a small category, but preserves the composition of functors _between_ small categories (ie composition of arrows of \\(\textbf{Cat}\\)).