Warning 7.6.5.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given an equalizer di Definition 7.6.5.11 (Equalizer and Coequalizer Diagrams). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An equalizer diagram in $\operatorname{\mathcal{C}}$ is a limit diagram $( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. A coequalizer diagram is a colimit diagram $( \bullet \rightrightarrows \bullet )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.
in $\operatorname{\mathcal{C}}$. Then the image of (7.66) 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.