Categories embedded one in another

Question

Let $\mathfrak{C},\mathfrak{D}$ be two categories such that there exist two embeddings (i.e. faithful functors injective on the objects - in general not full functors!) $F: \mathfrak{C} \longrightarrow \mathfrak{D}$ and $G: \mathfrak{D} \longrightarrow \mathfrak{C}$.

Can I deduce something on the two categories, such as for instance that they are equivalent or isomorphic?

My question concerns the metacategory of all categories and not a specific case of category. As I understand from this post: "Categorical" Schröder–Bernstein theorem?

the answer is no, because any group may be seen as a one-point category.

Now, assume for instance that $F$ is a full embedding. What can I say in this case? As a specific case, I take as $\mathfrak{C}$ the category whose objects are pairs of the form $(X,\mathcal{F})$, where $X$ is a set and $\mathcal{F}$ a collection of subsets of $X$, and whose hom-sets $Hom((X,\mathcal{F}),(Y,\mathcal{G}))$ consist of functions $f: \to Y$ whose direct image sends any member of $\mathcal{F}$ to a member of $\mathcal{G}$. It is quite easy to see that the correspondence associating with $(X,\mathcal{F})$ the pair $(X,\{\{A\}: \, A \in \mathcal{F}\}$ (and fixing morphisms) is a full embedding of $\mathfrak{C}$ in the category $\mathfrak{D}$ whose objects are pairs of the form $(X,\mathscr{F})$ (with $X$ a set and $\mathscr{F}$ a collection of set systems on $X$) and whose hom-sets $Hom((X,\mathscr{F}),(Y,\mathscr{G}))$ consist of functions $f:X \to Y$ such that the "direct image" $P^2(f)$ sends members of $\mathscr{F}$ t members of $\mathscr{G}$ (I will call $\mathfrak{D}$ the category of $2$-hypergraphs). Even in this case, it is possible to define a correspondence associating $(P(X),\mathscr{F})$ with the $2$-hypergraph $(X,\mathscr{F})$ (in essence $\mathscr{F}$ is a set system on $P(X)$), and sends a morphism of $\mathfrak{D}$ to its direct image, is simply an embedding.

May I say something in this case? Probabily, these functors do not yield neither an adjunction. Are there other functors working better than mine?

Answer 1

No, this is already false for categories with one object. For categories with one object we are looking for two non-isomorphic monoids which properly inject into each other and we can take, for example, the free group $F_2$ and the free group $F_3$. Groups appearing in such a counterexample must in particular properly inject into themselves so cannot be co-Hopfian.

We can even arrange for the two embeddings to be adjoint. For example, if $C$ is a concrete category with forgetful functor $U : C \to \text{Set}$, which is faithful by hypothesis, much of the time in practice the left adjoint $F : \text{Set} \to C$ is also faithful. This happens e.g. for groups or rings or modules. But $C$ is almost never equivalent to $\text{Set}$, e.g. if $C$ is groups or modules it has a zero object.