Clarifying the definition of the Dual Map

$T'$ maps a linear functional $\phi\in W'$ to the linear functional $(\phi \circ T)\in V'$.
$$T': W'\to V' \\ \phi \stackrel{T'}\longrightarrow \phi\circ T$$

To see that $\phi \circ T$ is a functional consider its input and output. The $T$ acts first and its input is an element of $V$. Then the $\phi$ acts and its output is an element of $\Bbb F$. So $\phi\circ T$ is a function $V\to \Bbb F$. Graphically, these functions look like:

$$\require{AMScd} \begin{CD} V @>T>> W @>\phi>> \Bbb F\\ @. \large\searrow @. \large\nearrow \\ @. \stackrel{\large\longrightarrow}{\phi\circ T} \end{CD}$$

Because compositions of linear maps are linear, this shows that $\phi\circ T$ is in fact an element of $V':=\mathcal L(V,\Bbb F)$.

It might be a bit easier to understand with different notation. If $T:V\to W$, then the dual map $T':W'\to V'$ satisfies $\phi[Tv]=(T'\phi)[v]$, for all $v\in V$ and $\phi\in W'$. That is, you can either first map $v$ to $Tv=w\in W$ and then apply $\phi$ to $w$, or first map $\phi$ to $T'\phi=\psi\in V'$ using $T'$ and then apply $\psi$ to $v$, getting the same scalar result either way.