---- > [!definition] Definition. ([[representable functor]]) > Let $\mathsf{C}$ be a (locally small) [[category]]. For each object $X$ of $\mathsf{C}$ let $\text{Hom}(X,-)$ and $\text{Hom}(-,X)$ be the covariant and contravariant [[hom functor|hom functors]] respectively. > A [[covariant functor]] $\mathscr{F}:\mathsf{C} \to \mathsf{Set}$ is said to be **representable** if $\mathscr{F}$ is [[natural transformation|naturally isomorphic]] to $\text{Hom}\big(A,-\big)$ for some object $A$ of $\mathsf{C}$. We say that $A$ **represents** $\mathscr{F}$. > A [[contravariant functor]] $\mathscr{G}:\mathsf{C} \to \mathsf{Set}$ (i.e., a [[covariant functor]] $\mathsf{C}^{\text{op}} \to \mathsf{Set}$) is representable if $\mathscr{G}$ is [[natural transformation|naturally isomorphic]] to $\text{Hom}(-,A)$ for some object $A$ of $\mathsf{C}$. We say that $A$ **represents** $\mathscr{G}$. ^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 > ```