Model theory preliminaries I'm reading this book, and as a preliminary I looked into the introduction to model theory included in the appendix B. On p. 625 (643 of the pdf), the authors write:

If $\mathbf M \subseteq \mathbf N$, then the inclusions $a \to a: M_s \to N_s$ yield an embedding $\mathbf M \to  \mathbf N$, called the natural inclusion of $\mathbf M$ into $\mathbf N$. Conversely, a morphism $h: \mathbf M \to \mathbf N$ yields a substructure $h(\mathbf M)$ of $\mathbf N$ whose underlying set of sort $s$ is $h_s(M_s)$

The definitions of a morphism and an embedding they use are provided in the text just above the quote.
My questions are:

*

*What is meant by “the inclusions $a \to a: M_s \to N_s$”? Specifically, I do not see a definition of an inclusion anywhere in the text. Do they mean just a mapping? If so, since it's $a \to a$ is it just an identity mapping? But then why isn't it called that?


*Where does "conversely" come from? I do not see how the statement that follows is a converse of the preceding sentence.


*Why is the last sentence true? It is not immediately obvious to me how to construct such a substructure.
 A: *

*The notation $\mathbf M\subseteq\mathbf N$ reads "$\mathbf M$ is included in $\mathbf N$" (or "$\mathbf M$ is a subset of $\mathbf N$"), and it means that every element of $M_s$ is an element of $N_s$, for every sort $s$. The inclusion map is the identity function from $M_s$ to $N_s$. For example, if $\mathbf M$ is the rationals, and $\mathbf N$ is the reals, then the inclusion map sends each rational number $\frac pq$ to the real number $\frac pq$.
My guess for why we don't call it the identity map, is that we usually think of the identity map having the same domain and range, while with an inclusion map the range is (often) a superset of the domain.


*If we have an inclusion, then we obtain a (canonical) embedding, in the form of the inclusion map. Conversely, if we have a morphism, then we obtain a substructure by considering the image of our morphism as a subset of the range. Inclusions give rise to morphisms (embeddings in fact), conversely morphisms give rise to inclusions.


*For each sort $s$, the morphism $h$ preserves sort, so the image $h_s[M_s]$ is included in $N_s$. Furthermore, $h$ is a morphism, thus it preserves relations and functions. Through induction over the complexity of $\mathcal L$-formulas, you can show that the range of $h$ is a substructure (basically, because of preserving relations and functions, the image $h[\mathbf M]$ behaves like $\mathbf M$, hence it is a structure, but $h[\mathbf M]$ is also a subset of $\mathbf N$, so it's a substructure of $\mathbf N$).
You're probably already familiar with this in a more applied setting. Take for example, a group homomorphism $h:\mathbf G\to \mathbf F$, then $h$ preserve the identity (which is a constant, or $0$-ary function) and the group operations (the $2$-ary function for the group addition, the $1$-ary function for the additive inverse), thus we can prove that the image $h[\mathbf G]$ is a group. But it's also included in the range $\mathbf F$, hence $h[\mathbf G]$ is a subgroup of $\mathbf F$.
