TITLE: $\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$
QUESTION [11 upvotes]: Introduction My Semester just started and we have a new Professor for Linear Algebra II (replacing our former Professor). Apparently we are behind our schedule and thus we only had a brief introduction (5 Minutes on the Blackboard) about the Dual Space and canonical isomorphisms.
In the current problem set there is this optional (not mandatory and not accredited) exercise which I understand as good as nothing about:

Problem: Let $V$ and $W$ be finite dimensional $\mathbb{K}$-Vectorspaces$^{1)}$. Show that $\hom(V,W)$ is canonical isomorph to $\hom(W^*,V^*)$ and find the canonical isomorphism. 
$^{1)}$ ($\mathbb{K}$ denotes a field here, I presume in english literature they write $\mathbb{F}$)  

Hint (given by my tutor): $$ \Phi: \hom(V,W) \longrightarrow  \hom(W^*,V^*) \\ \Psi \longmapsto (\varphi \longmapsto \varphi\circ \Psi)$$
My problems are vast now:

The literature I am reading does a bad job at explaining this topic
The online literature I find seems to be way beyond my level, they usually include tensor algebra to solve this. So I fall down the math rabbit hole see, On "familiarity" (or How to avoid "going down the Math Rabbit Hole"?)
I don't understand the Hint, I did not encounter such a function before, not even in Analysis and I don't know what's happening. I can only guess that it is a function that somehow completes a circle. 

On the bright side: 

I believe to understand what a canonical isomorphism is. My Professor said it is an isomorphism that does not involve making a choice for a basis. In the above function we do not make such a decision and therefore it would be a candidate for a canonical isomorphism. 
If I would understand the above function, I'd merely have to show that it is injective (trivial Kern) and surjective to complete the task. Unfortunately, as far, we've only done such things using an explicit Matrix and not a function. 
I do know the definition of the Dual Space, also using the Kronecker Delta.  (although I know nothing about it's meaning and geometric intrepreation, if there is any) 
The way my tutors sell this exercise it's supposed to be very easy, once the definitions are clear. 

To-Do-List: 

Understand the function, know what's happening.
Show that the function is surjective   
Show that the function is injective  (trivial Kern)

REPLY [4 votes]: IMO, the only hard part about this problem is wrapping your head around the notion of a function-valued function. Or worse, this problem is about a function-valued-function-valued function of functions.
When I first started learning about these things, I managed by writing with a very precise syntax, so I could see exactly what type everything is, and exactly what substitutions I could make, and I would introduce variables so that I could work pointwise as much as possible.
In this case, we can introduce variables as

$\Psi$ is a variable denoting a map $V \to W$
$\varphi$ is a variable denoting a functional on $W$
$v$ is a variable denoting a vector on $v$

and we can type things as:

$\Phi : \hom(V, W) \to \hom(W^*, V*)$
$\Phi(\Psi) : W^* \to V^*$
$\Phi(\Psi)(\varphi) \in V^*$
$\Phi(\Psi)(\varphi)(v) \in \mathbf{K}$.

The definition you cited is
$$ \Phi(\Psi)(\varphi)(v) := \varphi(\Psi(v)) $$
Once you're used to these things, it turns out this is "obvious", since the right hand side is pretty much the only way you can combine the three symbols $\varphi$, $\Psi$, and $v$ in the same expression, without mixing in other stuff.
Now, for example, to show that $\Phi$ is additive, we need to show
$$ \Phi(\Psi + \Psi')  = \Phi(\Psi) + \Phi(\Psi') $$
but since equality and addition of functions is defined pointwise, this is the same thing as showing 
$$ \Phi(\Psi + \Psi')(\varphi)(v)  = \Phi(\Psi)(\varphi)(v) + \Phi(\Psi')(\varphi)(v) $$
which can be seen after making the substitution.
Once you're comfortable doing everything pointwise, you can start thinking in terms of the functions themselves... but every now and then I still find it useful to go back to this approach to things when problems get complicated.
Of course, you weren't asked to show that $\Phi$ was additive, but it is, and it's a good thing to show. Everything else you might want to prove goes in a similar way.