Lemma 2.3.3 from Marker:
Suppose $\mathcal N$ is an $\mathcal L_M$-structure and $\mathcal N\models \operatorname{Diag}( \mathcal M)$; then, viewing $\mathcal N$ as an $\mathcal N$-structure, there is an $\mathcal L$-embedding of $\mathcal M$ into $\mathcal N$.
First of all, I assume $\mathcal N\models \operatorname{Diag}( \mathcal M)$ means $\mathcal N\models \phi(m_1,\dots,\phi_n)$ for all $\phi(m_1,\dots,m_n)\in \operatorname{Diag}( \mathcal M)$.
I have a few question about the proof.
Define $j:M\to N$ by $m\mapsto m^\mathcal N$.
Injectivity. If $m_1,m_2\in M$ are distinct, then $(m_1\ne m_2) \in \operatorname{Diag}( \mathcal M)$. Why is $j(m_1)\ne j(m_2)$? Here is my conjecture: if $j(m_1)= j(m_2)$, then $m_1^\mathcal N=m_2^\mathcal N$, which is the same as $m_1=m_2$ because $m_1,m_2\in M$ and any for any $m\in M$, $m^\mathcal N=m$. But if $m_1=m_2$ in $N$, then $\mathcal N\models (m_1=m_2)$, which is not true because $\mathcal N\models \operatorname{Diag}( \mathcal M)$ and $(m_1\ne m_2)\in \operatorname{Diag}( \mathcal M)$.
Is this reasoning correct? I used the assumption that $m^\mathcal N=m$ for all $m\in M$. But why is this true? According to this answer, if $\mathcal M$ is an $\mathcal L$-structure and we expand $\mathcal L$ by adding constants corresp. to elts of $M$, then $m^\mathcal M=m$. But in our case I'm using that $m^\mathcal N=m$, and we don't know the relationship between $\mathcal M$ and $\mathcal N$.
Function symbols are preserved. Here Marker writes that if $f$ is a function symbol in $\mathcal L$ and $f^\mathcal M(m_1,\dots,m_n)=m_{n+1}$, then $f(m_1,\dots,m_n)=m_{n+1}$ is a formula in $\operatorname{Diag}( \mathcal M)$. But why so? We don't know that $f^\mathcal M$ is $f$, do we? Further, Marker says that $f^\mathcal N(j(m_1),\dots,j(m_n))=j(m_{n+1})$. If what I wrote above (namely $m^\mathcal N=m$) is true, then $j(m_i)=m_i$, but we don't know anything about $f^\mathcal N$, do we?
Relation symbols are preserved. Let $R$ be a relation symbol. We must show $m=(m_1,\dots,m_n)\in R^\mathcal M$ iff $(j(m_1),\dots,j(m_n))\in R^\mathcal N$. Marker says that if $m\in R^\mathcal M$ then $R(m)\in \operatorname{Diag}( \mathcal M)$ and $(j(m_1),\dots,j(m_n))\in R^\mathcal N$, without mentioning the other direction. But I can't seem to understand either direction...
Also, he doesn't say that the constant symbols are preserved, is it because constant symbols are 0-ary function symbols, and that function symbols are preserved have been verified already?
Sorry if my questions seem elementary to someone, I really have a hard time understanding all this.