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**slevvio** Hello, I am reading some notes on category theory, and I just had a question about the forgetful functor?

Consider the forgetful functor $\displaystyle F: \textbf{Gp} \rightarrow \textbf{Set}$. I was wondering if the structure $\displaystyle \tilde{\textbf{Gp}} $ where the collection of objects is $\displaystyle \{ F(G) | G \in \text{Ob}(\textbf{Gp})\}$ and the collection of morphisms between two objects $\displaystyle F(G)$ and $\displaystyle F(H) $ is $\displaystyle \{ F(\phi) | \phi \in \text{Hom}_{\textbf{Gp}}(G,H))\}$ forms a category?

The reason I ask is because I read that $\displaystyle \textbf{Gp}$ was a subcategory of $\displaystyle \textbf{Set}$ but the objects of groups arent really contained in the objects of Sets, but I can see it if this 'image' of $\displaystyle \text{Gp}$ is a category. Thanks very much for any advice.