A subcategory of $\mathcal{C}$ is a category $\mathcal{A}$ satisfying the following conditions:
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1.
Objects. We have $\operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )\subset \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.
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2.
Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )$, we have
\[ \operatorname {\mathrm{Hom}}_{\mathcal{A}}\webleft (A,B\webright ) \subset \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright ). \] -
3.
Identities. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )$, we have
\[ \mathbb {1}^{\mathcal{A}}_{A} = \mathbb {1}^{\mathcal{C}}_{A}. \] -
4.
Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{A}\webright )$, we have
\[ \circ ^{\mathcal{A}}_{A,B,C} = \circ ^{\mathcal{C}}_{A,B,C}. \]
