definition. full and faithful [kostecki2011introduction, 3.2] [tt-0017]

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A functor \(\mathscr {F}: {\cal C} \to {\cal D}\) is full iff for any pair of objects \(X, Y\) in \({\cal C}\) the induced map \(F_{X, Y}: {\cal C}(X, Y) \to {\cal D}(\mathscr {F}(X), \mathscr {F}(Y))\) is surjective (onto). \(\mathscr {F}\) is faithful if this map is injective (one-to-one).