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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