The above sreenshot is from Takesaki's book (Vol 2, Chapter VIII).
In the proof of Proposition 3.15. I met with some questions.
Question1: If we let $g(x)=\varphi(axb), h(x)=\psi(axb)$, we have $g(x)=h(x)$ for all $x\in m_0$. $m_0$ is not dense in $M$, how to conclude $g(x)=h(x)$ for all $x\in M$.
Question 2: why can we construct an increasing net in $m_0^{+}$ converging $\sigma$-strongly to 1. I think we can construct a net in $m_0$ which is $\sigma$-weakly covergent to 1 since 1 is in $m_{\varphi}$ and $m_0$ is a $\sigma$-weakly dense $*$-subalgebra of $m_{\varphi}$
It doesn't say it in the statement, but if you read the proof he is assuming that both weights are normal. That answers 1.
For 2, one way to argue is to instead the unital C$^*$-algebra $\overline{\mathfrak m_0}$. Let $\{a_j\}$ be an approximate unit. It is well-known that $a_j\nearrow1$ wot, which agrees with the $\sigma$-weakly convergence since the net is bounded. Working in the norm closure of $\mathfrak m_0$ does not hamper the proof, since everything is norm-continuous.