Dear Seamus, an example of non-isomorphic objects mutually monomomorphing into each other is the following, in the category of groups ( I haven't tried to follow your sketch of construction).

Consider the free group on two generators $F_2$. Its commutator subgroup $C\subset F_2$ is a free group on denumerably many generators: $C=F_\infty$. This can be proved elegantly by using topological covering spaces [you can look it up in Massey's Introduction to Algebraic Topology for example].

So you have monomorphisms $F_2 \hookrightarrow F_\infty$ and $F_\infty \hookrightarrow F_2$, although $F_2$ and $F_\infty$ are not isomorphic, since their abelianizations are free $\mathbb Z$ modules on respectively two and denumerably many generators.

I have used that monomomorphisms in the category of groups coincide with injective morphisms, which is a not trivial but true result [ Jacobson, Basic Algebra, vol.II, Prop 1.1]

Dear Seamus, an example of non-isomorphic objects mutually monomorphing into each other is the following, in the category of groups ( I haven't tried to follow your sketch of construction).

Consider the free group on two generators $F_2$. Its commutator subgroup $C\subset F_2$ is a free group on denumerably many generators: $C=F_\infty$. This can be proved elegantly by using topological covering spaces [you can look it up in Massey's Introduction to Algebraic Topology for example].

So you have monomorphisms $F_2 \hookrightarrow F_\infty$ and $F_\infty \hookrightarrow F_2$, although $F_2$ and $F_\infty$ are not isomorphic, since their abelianizations are free $\mathbb Z$ modules on respectively two and denumerably many generators.

I have used that monomorphisms in the category of groups coincide with injective morphisms, which is a not trivial but true result [ Jacobson, Basic Algebra, vol.II, Prop 1.1]