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}\$$).