---- > [!definition] Definition. ([[fully faithful functor]]) > Let $\mathsf{C}$ and $\mathsf{D}$ be (locally small) [[category|categories]] and $\mathscr{F}:\mathsf{C} \to \mathsf{D}$ a [[covariant functor|functor]]. For each pair $A,B$ of objects, $\mathscr{F}$ defines a function $\text{Hom}_{\mathsf{C}}(A,B) \to \text{Hom}_{\mathsf{D}} \big( \mathscr{F}(A), \mathscr{F}(B) \big);$ if this function is both [[injection|injective]] and [[surjection|surjective]] — i.e., if $\mathscr{F}$ is both [[faithful functor|faithful]] and [[full functor|full]], then we call $\mathscr{F}$ **fully faithful**. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```