Let $\mathscr{Y}$ be a connected regular flat proper $R$-scheme such that $\mathscr{Y}_{k}$ is a strict normal crossings divisor. Then for every connected flat proper $R$-scheme $\mathscr{Z}$ and every isomorphism of $R/(t^{2})$-schemes

$$f:\mathscr{Y}\times_{R}R/(t^{2})\to\mathscr{Z}\times_{R}R/(t^{2}),$$

the following properties hold.

1. (1)

   The scheme $\mathscr{Z}$ is regular and $\mathscr{Z}_{k}$ is a divisor with strict normal crossings.

2. (2)

   Denote by $S^{+}$ the log scheme associated to $R\setminus\{0\}\to R$ and by $\mathscr{Y}^{+}$ and $\mathscr{Z}^{+}$ the schemes $\mathscr{Y}$ and $\mathscr{Z}$ endowed with the divisorial log structures associated to their special fibers. For every integer $d>0$ we denote by $s^{+}_{d}$ the standard log point $(\mathrm{Spec}\,k,k^{*}\oplus\mathbb{N})$ viewed as a log scheme over $S^{+}$ via the morphism of charts $\mathbb{N}\to\mathbb{N}:n\mapsto dn$. If we denote by $e$ the least common multiple of the multiplicities of the components of $\mathscr{Y}_{k}$, then there exists an isomorphism of log schemes

   $$g:\mathscr{Y}^{+}\times_{S^{+}}s^{+}_{e}\to\mathscr{Z}^{+}\times_{S^{+}}s^{+}_{e}$$

   over $s^{+}_{e}$, such that $g$ is compatible with the reduction of $f$ modulo $t$ (meaning that the obvious square in the category of $k$-schemes commutes).