Preorder Induced by Preorder Category

Theorem
Let $\left({S, \precsim}\right)$ be a preordered set.

Let $\mathbf S$ be its associated preorder category.

Let $\precsim'$ be the preorder induced by $\mathbf S$ as on Category Induces Preorder.

Then $\precsim'$ is the same as $\precsim$.

Proof
Suppose that for some $a, b \in S$, we have:


 * $a \precsim' b$

By Category Induces Preorder, this happens iff there exists an $\mathbf S$-morphism $f: a \to b$.

By definition of $\mathbf S$, this $f: a \to b$ exists iff:


 * $a \precsim b$

Hence the result.