Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth transformation. Define the pullback $T^*: C^k (\mathbb{R}^m) \rightarrow C^k (\mathbb{R}^n)$ (With $C^k(\mathbb{R}^n)$ being the set of functionals on $k$-cells on $\mathbb{R}^n$) by $T^*: Y \mapsto Y \circ T$.
Thus, the pullback of $Y \in C^k (\mathbb{R}^m)$ is the functional on $C_k (\mathbb{R}^n)$ $T^*Y:\phi \mapsto Y(T\circ \phi)$
Why is the pullback linear, and why does $(T \circ S)^* = S^* \circ T^*$?