Definition 4.29.2. Let $\mathcal{C}$ be a $2$-category. A *sub $2$-category* $\mathcal{C}'$ of $\mathcal{C}$, is given by a subset $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ of $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and sub categories $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}'}(x, y)$ of the categories $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ for all $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ such that these, together with the operations $\circ $ (composition $1$-morphisms), $\circ $ (vertical composition $2$-morphisms), and $\star $ (horizontal composition) form a $2$-category.

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