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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