Slice Category of Order Category

Theorem
Let $\mathbf P$ be a poset category.

Let $p \in \mathbf P_0$ be an object of $\mathbf P$.

Then:


 * $\mathbf P / p \cong \mathop{\bar \downarrow} \left({p}\right)$

where:


 * $\mathbf P / p$ is the slice of $\mathbf P$ over $p$;
 * $\mathop{\bar \downarrow} \left({p}\right)$ is the poset category defined by the weak lower closure of $p$.