Does the forgetful functor from $\mathbf{VEC}_{\mathbf{R}}$ to $\mathbf{SET}$ have right adjoint? Let $\mathbf{SET}$ be a category of sets, and $\mathbf{VEC}_{K}$ be a category of vector spaces over a field $K$.
In my course on Category Theory we discussed the concept of adjoint functor.
For example, we have constructed a functor $F$ that is left adjoint to the forgetful functor from $\mathbf{VEC}_{K}$ to $\mathbf{SET}$. Indeed, it is enough to take a functor that sends every set $A$ to some vector space with a basis $A$.
But what about the right adjoint functor to this forgetful functor in the case of $K$ equal to the field of real numbers $\mathbf{R}$?
After several unsuccessful attempts to come up with such a functor, it began to seem to me that such a functor does not exist at all.
Is it so? If so, why? If not, then how to build such a functor?
Any hints or advices would really help me, thank you!

If the functor forgetful functor $F$ were to admit a right adjoint, then it would preserve initial objects.
Therefore, it would need to map the zero vector space to the empty set.
This is clearly not the case.
