Is the interpretation of a constant symbol an injective map?

Question

The context

Im trying to show that the reduct to the luanguage $\frak{L}$ of any model of the complete diagram $D(\frak{M})$ of an $\frak{L}$-structure $\frak{M}$ is an elementary extension of some isomorphic copy of $\frak{M}$.

recall that $D(\frak{M})=\mathrm{Th}(\frak{M}^*)$, where $\frak{M}^*$ is the expansion of $\frak{M}$ to the luanguage $\frak{L}_\mathrm{M}$(we add a constant symbol $c_m$ for each $m$ and we interpret it as $m$)

In the proof. they consider $\frak{N}'\models D(\frak{M})$, a model for the complete diagram and they claim that the map $m\mapsto c_m^{\frak{N}'}$ provides the isomorphism!!!

The Question

But for this to work we mustn't have two constant symbol with the same interpretation, right??

Do we always assume, when we say that $\frak{M}$ is an $\frak{L}-$structure, that $M$ has a cardinality greater or equal to the cardinality of the constant symbols? and that we don't interpret two different symbols as the same element?

Or does $\frak{N}'$ being a model of the complete diagram ensures somehow the injectivity of that map??

Answer 1

Or does $\mathfrak{N}'$ being a model of the complete diagram ensures somehow the injectivity of that map??

Yes, that's exactly what happens. Whenever $m,m'$ are distinct elements of $\mathfrak{M}$, the sentence $$c_m\not=c_{m'}$$ is in $Th(\mathfrak{M}^*)$. So if $\mathfrak{N}'\models Th(\mathfrak{M}^*)$, the map $m\mapsto c_m^{\mathfrak{N}'}$ is indeed injective. (This is why it's crucial that we build the relevant theory after adding the new constant symbols!)

That said, in general you need not have distinct constants name distinct elements.