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The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories. Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \psi$, ...

Let $C$ be a small (1-)category. There is always a poset $D$ and a functor $p : D \to C$ such that: $p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$,...