Dual Map as Composition

Consider the definition of dual map, $$T'$$ in Axler (2015).

If $$T\in\mathcal{L}(V,W)$$, then the dual map of $$T$$ is the linear map $$T'\in\mathcal{L}(W',V')$$ defined by $$T'(\varphi)=\varphi\circ T$$ for $$\varphi\in W'$$.

My Question:
(1) I don't understand how the composition $$\varphi\circ T$$ works. $$T$$ maps from $$V$$ to $$W$$. Then, how does $$\varphi\circ T$$ gives you a linear map from $$V$$ to $$\mathcal{F}$$ since $$\varphi$$ is an element of $$W'$$, how does this composition works?

Reference: Axler, Sheldon J. $$\textit{Linear Algebra Done Right}$$, New York: Springer, 2015.

$$W'$$ is the dual space of $$W$$: i.e. it is shorthand for $$\mathcal{L}(W,\mathcal{F})$$.
Thus $$\varphi \in W'$$ is a map $$W \rightarrow \mathcal{F}$$, and $$\varphi \circ T$$ is a map $$V \rightarrow W \rightarrow \mathcal{F}$$, which means it is an element of $$\mathcal{L}(V,\mathcal{F})$$ aka $$V'$$.
Thus $$T'$$, which takes an element $$\varphi \in W'$$ and returns $$\varphi \circ T \in V'$$, is an element of $$\mathcal{L}(W',V')$$
• Thanks for the response! A quick questions: how does $\varphi\in W'$ imply it is a map from $W$ to $\mathcal{F}$? – Frank Swanton Jul 15 '19 at 18:58
• @FrankSwanton That's what $W'$ means. (You should check to see where it was defined in your book.) – Chessanator Jul 15 '19 at 19:07