Let $Y\subset{\mathcal{X}}_{0}$ be a stratum corresponding to face $\sigma$ of $\Delta({\mathcal{X}})$, and denote by $J\subset I$ the set of irreducible components $E_{i}$ cutting out $Y$. Then

$$B^{\mathcal{L}}_{Y}:=\sum_{i\notin J}(1-(a_{i}-\kappa_{\min}b_{i}))E_{i}|_{Y}$$

is a ${\mathbb{Q}}$-divisor on $Y$ with snc support, and we have a canonical identification

$${\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}:=K_{Y}+B^{\mathcal{L}}_{Y}$$

as ${\mathbb{Q}}$-line bundles. If we further assume that $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$, then $B^{\mathcal{L}}_{Y}$ has coefficients $<1$, so the pair $(Y,B^{\mathcal{L}}_{Y})$ is subklt.