The Clowder Project

Let $\webleft (X,\preceq _{X}\webright )$ be a poset and let $\mathcal{C}$ be a category.

1. Functoriality. The assignment $\webleft (X,\preceq _{X}\webright )\mapsto X_{\mathsf{pos}}$ defines a functor
   \[ \webleft (-\webright )_{\mathsf{pos}}\colon \mathsf{Pos}\to \mathsf{Cats}. \]
2. Fully Faithfulness. The functor $\webleft (-\webright )_{\mathsf{pos}}$ of Item 1 is fully faithful.
3. Characterisations. The following conditions are equivalent:
   1. The category $\mathcal{C}$ is posetal.
   2. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each $f,g\in \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$, we have $f=g$.