ScalingStacks

Proof. To prove (1), let $k\in\map(x,y,z)$ be a composition of $f$ and $g$ which results in a composite $g\circ f$. To show that $(g\circ f)_{*}\sim g_{*}\circ f_{*}$, it suffices to show that both sides of the equation are equal (in the homotopy category of spaces) to the zig-zag

$$\map(w,x)\xrightarrow{\{k\}\times 1}\map(x,y,z)\times\map(w,x)\xleftarrow{\sim}\map(w,x,y,z)\rightarrow\map(w,z).$$

The proof that $(g\circ f)^{*}\sim f^{*}\circ g^{*}$ is similar.

The proof of (2) is straightforward. ∎