This is trivial, I'm surprised it took me this long to internalize this fact.

When we convert a poset $(X, \leq)$ into a category, we stipulate that $x \rightarrow y \iff x \leq y$.

If we now consider the category $Set$ of sets and functions between sets, and arrow $A \xrightarrow{f} B$ is a function from $A$ to $B$. If $f$ is monic, then we know that $|A| = |Im(f)| \leq |B|$. That is, a monic arrow behaves a lot like a poset arrow!

Similarly, an epic arrow behaves a lot like the arrow in the inverse poset. I wonder if quite a lot of category theoretic diagrams are clarified by thinking of monic and epic directly in terms of controlling sizes.