Let $\mathcal{F}$ and $\mathcal{F}^{\prime}$ be stitched fibrations. Then,

• (i)

  two stitched fibrations $\mathcal{F}$ and $\mathcal{F}^{\prime}$ are conjugate if and only if $\inv(\mathcal{F})=\inv(\mathcal{F}^{\prime})$;

• (ii)

  $\mathcal{F}$ is smooth if and only if $\inv(\mathcal{F})$ vanish;

• (iii)

  $\mathcal{F}$ becomes smooth after an admissible change of coordinates on the base if and only if $\inv(\mathcal{F})$ are fibrewise constant.