Suppose ${\mathcal{X}}$, ${\mathcal{X}}^{\prime}$ and ${\mathcal{X}}^{\prime\prime}$ are snc models, with ${\mathcal{X}}^{\prime}$ dominating ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime\prime}$ dominating ${\mathcal{X}}^{\prime}$.

• (a)

  If $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is surjective, then so is $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$.

• (b)

  If $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$ is injective and $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}}^{\prime})$ is surjective, then $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ is injective.

• (c)

  If $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ and $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}}^{\prime})$ are both surjective, then so is $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$.

• (d)

  If $r_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime}}\to\Delta({\mathcal{X}})$ and $r_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}^{\prime}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}}^{\prime})$ are both injective, then so is $r_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\colon A_{{\mathcal{X}}{\mathcal{X}}^{\prime\prime}}\to\Delta({\mathcal{X}})$.