The following definitions and proposition are taken from the paper The geometry of Frobenioids I by S. Mochizuki.
Let $\mathbb{N}_{\ge 1}$ be the set of positive integers. $\mathbb{N}_{\ge 1}$ is a directed set by the order relation $a|b$. Let $M$, $N$ be commutative monoids(the binary operations are written additively). A map $f\colon M \rightarrow N$ is called a morphism if $f(x + y) = f(x) + f(y)$ for any $x, y$ and $f(0) = 0$.
For every $n \in \mathbb{N}_{\ge 1}$, the multiplication map $x \rightarrow nx$ on $M$ is a morphism.
$M$ is said to be torsion-free if $nx = 0$ implies $x = 0$ for every $n \in \mathbb{N}_{\ge 1}$.
Let $M^{gp}$ be the groupification of $M$, i.e. the grothendieck group of $M$. Let $\psi\colon M \rightarrow M^{gp}$ be the canonical map. If $\psi$ is injective, $M$ is said to be integral.
$M$ is said to be saturated if $na \in \psi(M)$ for some $n \in \mathbb{N}_{\ge 1}$ and $a \in M^{gp}$, then $a \in \psi(M)$.
$M$ is said to be perfect if the multiplication map $x \rightarrow nx$ is bijective for every $n \in \mathbb{N}_{\ge 1}$.
The perfection $M^{pf}$ of $M$ is defined as follows. Let $M_a = M$ for every $a \in \mathbb{N}_{\ge 1}$. if $a|b$, we define a morphism $f_{ba}\colon M_a \rightarrow M_b$ by $f_{ba}(x) = (b/a)x$. Then $(M_a)_{a\in \mathbb{N}_{\ge 1}}$ and $(f_{ba})$ consitute a direct(or inductive) system. We define $M^{pf} = colim_{a\in \mathbb{N}_{\ge 1}} M_a$. There exists the canonical morphism $\phi\colon M = M_1 \rightarrow M^{pf}$.
My question: How do we prove the following proposition?
Proposition
(1) $\phi\colon M \rightarrow M^{pf}$ is injective if $M$ is torsion-free, integral, and saturated.
(2) $M$ is perfect if and only if $\phi$ is an isomorhism.
Motivation Recently(August, 2012), S. Mochizuki submitted a series of papers(Inter-universal Teichmuller Theory I,II,III,IV) which develops his new theory. As an application of his theory, he wrote a proof of ABC conjecture. I think the validity of the proof has not yet been confirmed by other mathematicians. However, considering his track record, I think it's worthwhile to read the papers. He referred to The geometry of Frobenioids I for the notation and terminolgy concerning monoids and categories.