definition. inclusion functor [leinster2016basic, 1.2.18] [tt-003S]

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Whenever \({\cal S}\) is a subcategory of a category \({\cal C}\) , there is an inclusion functor \(\mathscr {I} : {\cal S} \hookrightarrow {\cal C}\) defined by \(\mathscr {I}(S) = S\) and \(\mathscr {I}( f ) = f\) , i.e. it sends objects and arrows of \({\cal S}\) into themselves in category \({\cal C}\). It is automatically faithful, and it is full iff S is a full subcategory.