I'm trying to define a preordered set $(S, \prec )$ from small category $\mathcal{C}$ using slice categories $(A \downarrow \mathcal{C})$.
Yet I want to ask if anyone can see a flaw in my reasoning, please. My reasoning is the following.
Given category $\mathcal{C}$ consider slice category $(A \downarrow \mathcal{C})$ under object $A$. Now take the collection of all homsets in this slice category. By virtue of $\mathcal{C}$ being small, this collection is a set. By virtue of $\mathcal{C}$ being locally small, every homset is a set, hence we are able to define each homset's cardinality:
$|(A \downarrow \mathcal{C})(A,X)|$, for all $X \in |\mathcal{C}|$
I write this cardinality as $|(A,X)|$ for simplicity. So we can now define a preordered set $(S,\prec)$ as follows:
Take set $S$ to have elements the homsets of $(A \downarrow \mathcal{C})$ and say $X \prec Y $ whenever $|(A,X)| \leq |(A,Y)|$
It is clear that we have reflexivity and transitivity, right? the only property we lose from partial order $\leq$ is antisymmetry because homsets with the same cardinality need-not be equal.
Any feedback would be greatly appreciated.