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Totally stuck at categories (I'm self-studying them, reading lecture notes from the course). Found the following problem in exercises:

There is an evident forgetful functor $$U:\mathbf{Vect_k} \rightarrow \mathbf{Set}.$$ Show that this function 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?

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    $\begingroup$ Show that $F$ assigns to each set $X$ the vector space with $X$ as a basis. $\eta_X$ is then the insertion of these generators. $\endgroup$ – Stefan Hamcke Nov 11 '13 at 0:10
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    $\begingroup$ Just a comment: left adjoints to forgetful functors often go under the heading "free". So, with this in mind, as Stefan suggests, the "free vector space" on a set seems like a good option. $\endgroup$ – Alex Youcis Nov 11 '13 at 7:04
  • $\begingroup$ Also note that there are many equivalent characterizations of adjointness. The definition I learned used a natural bijection between $X(x,Ga)$ and $A(Fx,a)$, where $F:X→A$ and $G:A→X$. But I think in most cases one uses the characterization via the universal arrows $η_x:x→GFx$ for each object $x\in X$. Here you have to show that a map $f:x→Ga$ there's a unique map $f':Fx→a$ such that $Gf'∘η_x=f$. The object function $F$ can then be extended to a functor $X→A$ in a unique way such that $η:x→GFx$ is a natural transformation. $\endgroup$ – Stefan Hamcke Nov 11 '13 at 13:45
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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 $\bf A$.

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