Please excuse me if the question is too vague or uninteresting.

Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. Motivated by the equivalence of Dedekind-finiteness and finiteness for sets (assuming Choice), let us define that an object $A$ is $\mathcal{C}$-infinite if there is an object $A'$ in $\mathcal{C}$ corresponding to a proper subobject of $A$ and an isomorphism $f: A \to A'$ in $\mathcal{C}$.

In the category **Set**, we have the property that ($B$ is **Set**-infinite and $B \subseteq A) \implies A$ is **Set**-infinite. The analogous property does not hold (with this definition of infinity) over arbitrary categories. For instance, the additive group $\mathbb{Q}$ has $\mathbb{Z}$ as a proper subgroup, is finite in the category **Grp** (since any homomorphism $f: \mathbb{Q} \to \mathbb{Q}$ must actually be an isomorphism), but $\mathbb{Z}$ is **Grp**-infinite by $g:\mathbb{Z} \to 2\mathbb{Z}$ with $z \mapsto 2z$.

In which categories would a version of this transitivity property hold? What about other properties depending on a notion of infinity? Or determining whether a given object in a category is infinite or not?

A result somewhat related to this notion is that a Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space. Any such Banach space is obviously **Ban**-infinite.

I don't know much Category Theory or Logic, but this has been troubling me for a while (I'm an undergraduate student). Thanks in advance for your contributions.