$\textbf{Motivation:}$
Given two objects $A$ and $B$ in a category $\mathscr{C}$, we can compare them in different ways. For instance, $A \cong B$ tells us that there exists at least one isomorphism from $A$ to $B$. This relation is of course an equivalence on the class of objects of $\mathscr{C}$ (in fact it allows us to form the groupoid core of $\mathscr{C}$), but we can significantly weaken it, $i.e.$ form a superobject of $\cong$. To that end, let $A \triangleleft B$ signify that there exists a morphism $A \longrightarrow B$. This relation is precisely the preorder induced by any category; it is reflexive and transitive, however $A \triangleleft B$ and $B \triangleleft A$ doesn't tell us too much about the relationship between $A$ and $B$.
$\textbf{Situation:}$
We may then want to strengthen our preorder into something analogous to a partial order. A good place to start might be to define $A \precsim B$ to mean "$A$ is a retract of $B$". The most general interpretation of this statement is that there exist morphisms $A \stackrel{s}{\longrightarrow} B \stackrel{r}{\longrightarrow} A$ such that $r \circ s=id_A$, $i.e.$ a section-retraction pair. From here we can already observe that
$\hspace{5 cm} A \precsim B \implies (A \triangleleft B) \wedge (B \triangleleft A)$
The relation $\precsim$ also happens to be a preorder, although now if we suppose that $A \precsim B$ and simultaneously $B \precsim A$, a much more interesting scenario arises. That is, in addition to the aforementioned pair $A \stackrel{s}{\longrightarrow} B \stackrel{r}{\longrightarrow} A$, we have another pair $B \stackrel{s'}{\longrightarrow} A \stackrel{r'}{\longrightarrow} B$ such that $r' \circ s'=id_B$. Drawing all this out diagrammatically yields a slew of composite morphisms related to one another in various ways. For example, we can form the pair $A \stackrel{s' \circ s}{\longrightarrow} A \stackrel{r \circ r'}{\longrightarrow} A$ which is indeed another nontrivial factorization of $id_A$, but we already have the tautology $A \precsim A$, so this might just be a distraction. We also have the split idempotents $e:B \stackrel{r}{\longrightarrow} A \stackrel{s}{\longrightarrow} B$ and $e':A \stackrel{r'}{\longrightarrow} B \stackrel{s'}{\longrightarrow} A$ which in some sense "feel weakly equivalent" to the respective identities on $B$ and $A$.
$\textbf{Question:}$
Under what assumptions is $\precsim$ "weakly" antisymmetric? That means, when does $A \precsim B$ and $B \precsim A$ imply $A \cong B$? (Replace $\cong$ by a weaker equivalence if necessary.) Since here we speak the language of retracts, I suspect there would be more to say provided $\mathscr{C}$ came equipped with some notion of homotopy between morphisms (such as in a model category, but I'm not too familiar with those). From this illuminating question I have gathered, that contrary to my intuition, objects/spaces which are each other's retracts not only do not need to be isomorphic, but aren't even always homotopy equivalent! So there might in fact exist a stronger relation between objects in $\mathscr{C}$ (with appropriate structure) which actually is weakly antisymmetric, without having to do the standard method of taking quotients (and hopefully without invoking the Axiom of Choice somehow).
Generally, is there some way of making better sense of the situation described above or am I dealing with a red herring? Are there any interesting examples/counterexamples which might help shed light on my somewhat broad question? I'm sorry if I'm getting a bit rambly; at the very least I would like to take this as an opportunity to learn more homotopy theory. Thanks for any and all insight offered.
$\textbf{Edit:}$ I've just properly noticed the connection between my main question and the Cantor-Bernstein theorem; indeed $\precsim$ is weakly antisymmetric within $\mathsf{Set}$ by the aforementioned theorem. Thus my question could then be rephrased as "Is there a particular class of categories in which the generalized Cantor-Bernstein theorem holds?", however this perspective makes the whole problem sound much scarier. Although being aware that Cantor-Bernstein implies the Law of Excluded Middle, I think it's safe to assume that the generalized version won't hold internally in a topos.