Remark 4.1.2.1. Let $\operatorname{\mathcal{C}}$ be a category. We will say that a category $\operatorname{\mathcal{C}}'$ is a subcategory of $\operatorname{\mathcal{C}}$ if it arises from the construction described above (for some collection of objects $\operatorname{Ob}'(\operatorname{\mathcal{C}})$ and collections of morphisms $\{ \operatorname{Hom}'_{\operatorname{\mathcal{C}}}(X,Y) \} _{X,Y \in \operatorname{Ob}'(\operatorname{\mathcal{C}})}$). Phrased differently, a category $\operatorname{\mathcal{C}}'$ is a subcategory of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
The set of objects $\operatorname{Ob}(\operatorname{\mathcal{C}}')$ is a subset of the set of objects $\operatorname{Ob}(\operatorname{\mathcal{C}})$.
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}') \subseteq \operatorname{Ob}(\operatorname{\mathcal{C}})$, the set of morphisms $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y)$ is a subset of the set of morphisms $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.
There is a functor $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ which is the identity on objects and morphisms.
We write $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ to indicate that $\operatorname{\mathcal{C}}'$ is a subcategory of $\operatorname{\mathcal{C}}$.