Elementary topos embeddings

In order theory, it is customary to embed posets in boolean algebras. It would be nice if topoi could play the role of boolean algebras in category theory: then preorders, monoids, and other categories can be embedded in elementary topoi. This is a vast generalisation of the representation of posets by set systems in order theory.

Order theory:
There are two fundamental ways of representing posets by embedding in $Sets$: (1) each element $x$ can be represented as a singleton sets $\{x\}$ with each $x \subseteq y$ represented by the unique function of singletons: $\{x\} \to \{y\}$ and (2) given a set theoretic representation (such as representation by principal ideals) each element can be represented by a set and each edge can be represented by the unique corresponding inclusion function.

Monoid theory:
Every monoid can be represented by its self-induced actions. Let $M$ be a monoid, then its left action $l : M \to Sets$ defined by $l_x(y) = xy$ is clearly a monoid action, and it is faithful because $M$ is a monoid. Self-induced actions on a semigroup don't need be faithful. As this equally well applies to groups, this reproduces Cayley's theorem from group theory.

Category theory:
The Yoneda embedding in category theory just generalizes the representation by self-induced actions in monoid theory. The interesting thing is this embedding $h : C \to [C^{op}, Sets]$ embeds any small category $C$ into it a set-valued functor topos, so it is safe to say that small categories are simply subcategories of elementary topoi. There are now two topos theoretic aspects of categories: (1) topos embeddings and (2) topos constituents of a category.

References:
[1] Yoneda embedding