Addendum to “Refinement invariance of intersection (co)homologies”

In a previous work we proved the refinement invariance of several intersection (co)homologies existing in the literature. Specifically, we worked with a refinement $f : (X, \mathcal{S}) \to (X,\mathcal{T})$ between two CS‑sets where the strata of $\mathcal{S}$ were embedded in the strata of $\mathcal{T}$. However, in this paper, we establish that this embedding condition is not a requirement for the refinement invariance property.