ScalingStacks

Proof. Let $U^{\prime}=X^{\prime}-(X^{\prime}_{f}\cup X^{\prime}_{exc}\cup\partial X^{\prime})$, a dense subset of $X^{\prime}$. Note that $U=f^{-1}(U^{\prime})$ is a dense subset of $X-(X_{exc}\cup\partial X)$. Given $x^{\prime}\in U^{\prime}$, fix a morphism $l_{x^{\prime}}:{\mathbf{Z}}^{C(x^{\prime})}\to{\mathbf{Z}}$ that does not factor through the sum map. Given $x\in U$, let $l_{x}=l_{x^{\prime}}\circ f:{\mathbf{Z}}^{C(x)}\to{\mathbf{Z}}$. Lemma 7.1.23 shows that $(l_{x}\circ(m_{c})_{c\in C(x)})_{x\in U}:R(X)\to{\mathbf{Z}}^{U}$ is injective. This map is equal to the composition

$$R(X)\xrightarrow{f}R(X^{\prime})\xrightarrow{(l_{x^{\prime}}\circ(m_{c^{\prime}})_{c^{\prime}\in C(x^{\prime})})_{x^{\prime}\in U^{\prime}}}{\mathbf{Z}}^{U^{\prime}}\xrightarrow{f^{*}}{\mathbf{Z}}^{U}$$

since $m_{f(c)}^{\pm}(f(\zeta))=m_{c}^{\pm}(\zeta)$ and $m_{f(c)}(f(\zeta))=m_{c}(\zeta)$ for all $x\in X$, $c\in C(X)$ and all homotopy classes of paths $\zeta$ in $X$ (Lemma 7.1.24). Since $H$ is contained in the kernel of the composition

$$R(X^{\prime})\xrightarrow{(l_{x^{\prime}}\circ(m_{c^{\prime}})_{c^{\prime}\in C(x^{\prime})})_{x^{\prime}\in U^{\prime}}}{\mathbf{Z}}^{U^{\prime}}\xrightarrow{f^{*}}{\mathbf{Z}}^{U},$$

it follows that the composite map of the lemma is injective. ∎