---- > [!definition] Definition. ([[universal element]]) > A **universal element** of a [[covariant functor|functor]] $\mathscr{F}:\mathsf{C} \to \mathsf{Set}$ is an element $\theta \in \mathscr{F}(X)$, where $X$ is some object of $\mathsf{C}$, which exhibits [[representable functor|representability]] of $X$ via [[the Yoneda lemma]]. That is, [[the Yoneda lemma]] says that $\theta \in \mathscr{F}(X)$ induces, in a [[natural transformation|natural]] [[bijection|bijective]] fashion, a [[natural transformation]] $\begin{align} \hat{\theta}: \text{Hom}\big(X, -\big)& \Rightarrow \mathscr{F} \\ \hat{\theta}_{Y}(f:X \to Y)& := \mathscr{F}(f)(\theta); \end{align}$ and we say $\theta$ is universal if $\hat{\theta}$ is in fact a [[natural transformation|natural isomorphism]]. > >The corresponding definition for a [[contravariant functor]] $\mathscr{G}$ is evident. ^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 > ```