Left adjoint of a forgetful functor Totally stuck at categories (I'm self-studying them, reading lecture notes from the course). I found the following problem in exercises:

There is an evident forgetful functor
$$U:\mathbf{Vect_k} \rightarrow \mathbf{Set}.$$
Show that this functor has a left-adjoint $F$ with a unit
$$\eta_X : X \rightarrow UF(X).$$
What property does the image of $\eta_X$ have within the vector space $F(X)$?

Do not really know where to start. Can you help please?
 A: You may read $U$ as the 'underlying set' functor, and $F$ as the 'free' functor. 
Note that the forgetful functor always has a left adjoint for each category of (all algebraic structures of the same kind).
In the case of vector spaces over $k$, the free vector space on a set $X$ is just a vector space of dimension $|X|$: we start out of the set $X$ and extend it by applying freely (meaning: only 'formally') the given operations on the arising elements. So that, in the end, we arrive to the free vector space $F(X)$ of formal linear combinations of elements of $X$, which means that $X$ (mapped into the underlying set $UF(X)$ of $F(X)$ via $\eta_X$) is a basis in $F(X)$.
Remark. I like to permit arrows between objects of two distinct categories $\bf A$ and $\bf B$ (so called 'heteromorphisms'), which together with the original arrows form a bigger category. In this case, we could define this bigger category as the disjoint union of ${\bf Set}$ and ${\bf Vect}_k$ plus all functions $A\to U(V)$ as heteromorphisms. 
Then, adjunction can be rephrased as

Each object of $\bf A$ has a reflection in $\bf B$, and each object of $\bf B$ has a coreflection in $\bf A$.

