Warning 7.6.4.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given an equalizer diagram
7.70
\begin{equation} \begin{gathered}\label{equation:equalizer-not-in-homotopy} \xymatrix@C =50pt@R=50pt{ Z \ar [r]^-{g} & Y \ar@ <.4ex>[r]^-{f_0} \ar@ <-.4ex>[r]_-{f_1} & X } \end{gathered} \end{equation}
in $\operatorname{\mathcal{C}}$. Then the image of (7.70) in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ need not be an equalizer diagram. In other words, the forgetful functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}})$ does not preserve equalizer diagrams in general.