Kerodon

Definition 7.6.5.4 (Equalizers and Coequalizers). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0, f_1: Y \rightarrow X$ be morphisms of $\operatorname{\mathcal{C}}$ having the same source and target, which we identify with functor $\sigma : ( \bullet \rightrightarrows \bullet ) \rightarrow \operatorname{\mathcal{C}}$. An equalizer of $f_0$ and $f_1$ is a limit of the diagram $\sigma $. A coequalizer of $f_0$ and $f_1$ is a colimit of the diagram $\sigma $. We say that the $\infty $-category $\operatorname{\mathcal{C}}$ admits equalizers if every pair of morphisms $f_0, f_1: Y \rightarrow X$ have an equalizer in $\operatorname{\mathcal{C}}$, and that $\operatorname{\mathcal{C}}$ admits coequalizers if every pair of morphisms $f_0, f_1: Y \rightarrow X$ have a coequalizer in $\operatorname{\mathcal{C}}$.