Exercise 10.2.2.16. Let $\operatorname{\mathcal{C}}$ be the category of partially ordered sets (where morphisms are nondecreasing functions). Let $Q = \{ a, b, c, d \} $ be a set with four elements, endowed with the partial ordering indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ a \ar [r] & b & c \ar [l] \ar [r] & d. } \]

Let $f: Q \rightarrow [2] = \{ 0 < 1 < 2 \} $ be the nondecreasing function given by

\[ f(a) = 0 \quad \quad f(b) = 1 = f(c) \quad \quad f(d) = 2, \]

so that we have a pullback diagram of partially ordered sets

10.21

\begin{equation} \begin{gathered}\label{equation:poset-not-universal} \xymatrix@R =50pt@C=50pt{ \{ a, d \} \ar [r] \ar [d]^{f_0} & Q \ar [d]^{f} \\ \{ 0 < 2 \} \ar [r] & [2]. } \end{gathered} \end{equation}

Show that $f$ is a quotient morphism in (the nerve of) the category $\operatorname{\mathcal{C}}$, but that $f_0$ is not.