ScalingStacks

Proof. This follows from the diagram

Here the vertical column is a weak equivalence since $f$ is a homotopy equivalence (restricting to the fiber over $f\in\map(x,y)_{f}$ of the projections to $\map(x,y)_{f}$ gives exactly the zig-zag which defines $f^{*}\colon\map(y,z)_{g}\rightarrow\map(x,z)_{g\circ f}$). Since $(d_{0},d_{2})$ is a weak equivalence, the lemma follows. ∎