# How to connect monoidal fans (Kato) to fans (Oda).

In the paper Toric Singularities, Kato defines log regular for a sheaf of monoids on a scheme. In section 9, he defines a fan in terms of monoidal spaces (and later that a log regular structure induces such a fan), and in 9.5 connects his concept of a fan to that given by Oda:

Let $L$ be a finitely generated free abelain group. A fan in $L$ in the sense of Oda is equivalent to a fan $F$ given by Kato endowed with a homomorphism of sheaves $h : \text{Hom}(L,\mathbb{Z}) \rightarrow M_F^\text{gp}$ which satisfies three conditions. The last condition is that the map $\text{Mor}(\text{Spec}(\mathbb{N}),F) \rightarrow L$ induced by $h$ is injective (here $\mathbb{N}$ is a monoid under addition).

How does $h$ induce such a map?

-