Preorder as a monoid object

Consider the category of endorelations on a set $X$$X$. Specifically, let $P(X\times X)$$P(X \times X)$ form a partial order category where the objects are elements of $P(X\times X)$$P(X \times X)$ (i.e., relations on $X$$X$), and the morphisms are inclusions. We denote this category as $\underset{―}{X}$$\underline{X}$.

This category admits a natural monoidal structure. The monoidal product is defined as relational composition, with the unit element being the diagonal relation $\mathrm{\Delta }$$\Delta$.

We observe that a preorder on $X$$X$ is precisely a monoid object in $(\underset{―}{X},\circ ,\mathrm{\Delta })$$(\underline{X}, \circ, \Delta)$.

The reason is straightforward: a relation $R$$R$ is transitive if and only if $R\circ R\subseteq R$$R \circ R \subseteq R$, and a relation is reflexive if and only if $\mathrm{\Delta }\subseteq R$$\Delta \subseteq R$.

The remaining associativity and unitality laws are evident. Since we are working in a partial order category, any diagram one can draw is commutative, and this forms a monoidal structure.

In addition, we have

$$\mathsf{Sym}(X)=\mathsf{Pic}(\underset{―}{X})$$