Let $\mathscr{A}$ be a category. A subcategory $\mathscr{S}$ of $\mathscr{A}$ consists of a subclass $\ob(\mathscr{S})$ of $\ob(\mathscr{A})$ together with, for each $S, S’ \in \ob(\mathscr{S})$, a subclass $\mathscr{S}(S, S’)$ of $\mathscr{A}(S, S’)$, such that $\mathscr{S}$ is closed under composition and identities. It is a full subcategory if $\mathscr{S}(S, S’) = \mathscr{A}(S, S’)$ for all $S, S’ \in \ob(\mathscr{S})$.