A subcategory of $\mathcal{C}$ is a category $\mathcal{A}$ satisfying the following conditions:
- Objects. We have $\text{Obj}\webleft (\mathcal{A}\webright )\subset \text{Obj}\webleft (\mathcal{C}\webright )$.
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Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{A}\webright )$, we have
\[ \textup{Hom}_{\mathcal{A}}\webleft (A,B\webright ) \subset \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ). \]
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Identities. For each $A\in \text{Obj}\webleft (\mathcal{A}\webright )$, we have
\[ \mathbb {1}^{\mathcal{A}}_{A} = \mathbb {1}^{\mathcal{C}}_{A}. \]
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Composition. For each $A,B,C\in \text{Obj}\webleft (\mathcal{A}\webright )$, we have
\[ \circ ^{\mathcal{A}}_{A,B,C} = \circ ^{\mathcal{C}}_{A,B,C}. \]
