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Exercise 10.2.2.15. Let $\operatorname{\mathcal{C}}$ be the (nerve of the) ordinary category depicted informally by the diagram

\[ \xymatrix@R =50pt@C=50pt{ \widetilde{X} \ar@ <-.4ex>[d]_{ e_0 } \ar@ <.4ex>[d]^{e_1} \ar [dr] & \widetilde{Y} \ar@ <-.4ex>[d]_{ g_0 } \ar@ <.4ex>[d]^{g_1} \ar [dr] & \\ X \ar [r]^-{ f } & Y \ar [r]^-{h} & Z, } \]

so that $f \circ e_0 = f \circ e_1$ and $h \circ g_0 = h \circ g_1$. Show that $f$ and $h$ are quotient morphisms in $\operatorname{\mathcal{C}}$, but the composition $(h \circ f): X \rightarrow Z$ is not a quotient morphism.