Definition
Let \(f:X\to Y\) be a morphism of log schemes. We say that \(f\) is strict if the map
\begin{align*} f^\flat:f^{-1}\mathcal M_{Y} \to \mathcal M_X \end{align*}induces an isomorphism of log structures (that is, an isomorphism of sheaves of monoids) between the pullback of the log structure on \(Y\) to \(X\) and the log structure \(\mathcal M_X\) on \(X\).