Given our one-object category, does the functor \$$F\$$ always produce a square out of the elements of the chosen set?

What happens if we change base to \$$\mathbf{Cost}\$$? Do we get a square with sides that have a specific length?

Such squares and other simple geometric shapes and their properties were studied by the ancient Greeks extensively. Can the mathematics of the ancient Greeks be defined as a functor or \$$\mathcal{V}\$$-functor?