definition. (full) subcategory [leinster2016basic, 1.2.18] [tt-002Y]

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Let \({\cal C}\) be a category. A subcategory \({\cal S}\) of \({\cal C}\) consists of a subclass \(\operatorname {Ob}({\cal S})\) of \(\operatorname {Ob}({\cal C})\) together with, for each \(S, S' \in \operatorname {Ob}({\cal S})\), a subclass \({\cal S}\left (S, S'\right )\) of \({\cal C}\left (S, S'\right )\), such that \({\cal S}\) is closed under composition and identities.

It is a full subcategory if \({\cal S}\left (S, S'\right )={\cal C}\left (S, S'\right )\) for all \(S, S' \in \operatorname {Ob}({\cal S})\).