Pos,coequalizer,terminal poset

Question

Consider the reflexive pair $u, v \colon (1 + 2) \to 2$ from the coproduct of the terminal poset $1$ and the two-element chain, where both morphisms are the identity on the second summand, and they map the first one to the top and bottom of $2$, respectively. Why is the coequalizer of $u$ and $v$ the terminal poset in $\mathsf{Pos}$ (the category of posets) ?

Answer 1

Hint. Take a morphism $f \colon 2 \to P$ in $\mathsf{Pos}$ such that $fu=fv$. What does it tell you about the image by $f$ of the top and bottom of $2$ ?