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

Consider the forgetful functor

. I was wondering if the structure

where the collection of objects is

and the collection of morphisms between two objects

and

is

forms a category?

The reason I ask is because I read that

was a subcategory of

but the objects of groups arent really contained in the objects of Sets, but I can see it if this 'image' of

is a category. Thanks very much for any advice.