Given a category $\mathcal{C}$, I was thinking about a subcategory $\mathcal{S}$ formed by throwing away all the morphisms having no left inverse(or right inverse). Then we denote, for each pair $X, Y \in \mathfrak{ob}(\mathcal{S})$, $X \leq Y$ if $\mathrm{Hom}_{\mathcal{S}}(X,Y)$ is not the empty class. Let's say $X \sim Y$ if $X \leq Y$ and $Y \leq X$. If we let $\mathcal{P}$ be a full subcategory of $\mathcal{S}$ consisting of each objects in the equivalance class, then we could define many order-(or set-)theoretic notions on $\mathcal{P}$.
My question is, what types of categories have this property: $X \sim Y$ if and only if $X$ and $Y$ are isomorphic as usual categorical sense? If a category satisfy this condition, then we can translate all the theory of poset on that category, so this question is extremely meaningful to ask.
Such examples include the the category $\mathbf{Set}$(Cantor-Bernstein Theorem), the category of finite dimensional vector spaces over a fixed field, etc. The $\mathcal{P}$ in the first case gives quite complicated model and the second case gives the structure isomorphic to the natural numbers.
My aim is to consider filters on this kind of poset and build a class of ultrafilters each of which representing a point of the category $\mathcal{P}$, analogous to the Stone–Čech compactification of a topological space. Although this procedure can be done with any category, I'm wondering the above question for the sake of computability of the points of the compactification of the category $\mathcal{P}$.
ps) Since I paused studying mathematics for a while and I don't have firm knowledge on the set theory dealing with classes, if there is any helpful material dealing with general foundation and construction of a set theory other than $ZFC$, please let me know. Thanks.