One of the motivating example of natural transformation in category theory is the canonical injective linear map from $V$ to $V^{**}$. I am trying to understand this example from scratch, but failing at some places. The natural injection from $V$ to $V^{**}$ is given by $v\mapsto e_v$, where $e_v$ is a member of $V^{**}$ which evaluates $f\in V^*$ at $v$.
Thus, if $\mathcal{V}$ denotes category of $k$-vector spaces, we have a map from $\mathcal{V}$ to $\mathcal{V}$, which takes an object $V$ to object $V^{**}$.
The next thing I am facing trouble in understanding is, how we define images of morphisms under this map (which is going to be double dual functor), i.e. given $T:V\rightarrow W$, a morphism in $\mathcal{V}$, I was facing problem in precisely defining $T^{**}:V^{**}\rightarrow W^{**}$. Now this seems very silly question, but, I was unable to write it. (After this, I can try to prove basic properties of functior for this "double dual map", and at last to prove that it is isomorphic to identity functor.)
Can one help me to write the induced map $T^{**}:V^{**}\rightarrow W^{**}$?
It's easier to define $T^*:W^*\to V^*$. Indeed, $T^*(f)(v)=f(T(v))$. Then iterate: $T^{**}(\xi)(f)=\xi(T^*(f))=\xi(v\mapsto f(T(v)))$. Here $f\in W^*$ and $\xi\in V^{**}$.