Definition 4.2.10. A subcategory of a category \mathcal{B} is a category \mathcal{A} whose objects and arrows form subsets of the objects and arrows of \mathcal{B} and such that source, target and composition in \mathcal{A} agree with those of \mathcal{B} and such that the identity morphism of an object of \mathcal{A} matches the one in \mathcal{B}. We say \mathcal{A} is a full subcategory of \mathcal{B} if \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, y) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(x, y) for all x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}). We say \mathcal{A} is a strictly full subcategory of \mathcal{B} if it is a full subcategory and given x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) any object of \mathcal{B} which is isomorphic to x is also in \mathcal{A}.
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