The common and central (to category theory) notion of <I>universal element</I> has at least four definitions.
Here we aim to present and compare the definitions.
They all assume we start with a functor
\[ \boxed{ \rightadj{ \leftcat{(\Set = \calL)} \longleftarrow \rightcat\calR } : R } \, . \]
Here the designations of what is "left" and what is "right"
is determined by
what is on the left and right within the hom sets of each category in the definition of adjunction (or representable).
In the first two definitions there are certain (hopefully) familiar ${\rightadj\forall}{\leftadj{\exists!}}$ conditions that form part of the definition;
I am not going to repeat those conditions -- you can look them up,
or read about them later in this blog
In the last two definitions those conditions are built into
the definitions of "initial object" and "left lift".
The rightmost columns are works in progress,
dual to the column next to them.
\[ \boxed{ \begin{array} {cll|ccccc|ccccc|ccccc} 1. & \text{universal element} & \leftadj{ \lambda_0 \in \rightcat{r_0} \rightadj R } \\ \hline 2. & \text{universal arrow} & \leftadj{\lambda_0 \in \leftcat{ \hom 1 \Set { \rightcat{r_0} \rightadj R } } } \text{, i.e.,} \\ && \leftadj { \lambda_0 : \leftcat{1} \to \rightcat{r_0} \rightadj R } \text{ in } \leftcat\Set \\ \hline &&& {} \rlap{ \kern2em \target{ \text{covariant form} } } &&&&& {} \rlap{ \kern8em \source{ \text{contravariant form} } } \\ &&& {} \rlap{ \kern1em \leftadj{ \text{universal element} } } &&&&& {} \rlap{ \kern8em \rightadj{ \text{opuniversal element} } } \\ \hline &&& \leftcat{ \calI \rlap{ \kern.5em \xleftarrow [\kern11em] {!} } } &&&& \leftcat{1 \downarrow {\rightadj R} } & \rightcat{ {\leftadj L} \downarrow 1 \rlap{ \kern.5em \xrightarrow [\kern11em] {!} } } &&&& \rightcat{\calI} & \leftcat{ 1 \downarrow {\rightadj L} \rlap{ \kern.5em \xrightarrow [\kern10em] {!} } } &&&& \rightcat{\calI} \\ 3. & \text{initial object} & \leftadj{r_0 \langle \lambda_0\rangle} & \leftcat{ \llap 1 \big\downarrow } && \smash{ \leftcat{ \stackrel \lambda \Longrightarrow } } && \rightcat{\big\downarrow \rlap r } & \leftcat{\llap l \big\downarrow } && \smash{ \rightcat{ \stackrel \rho \Longrightarrow } } && \rightcat{ \big\downarrow \rlap 1 } & \leftcat{\llap l \big\downarrow } && \smash{ \rightcat{ \stackrel \rho \Longleftarrow } } && \rightcat{ \big\downarrow \rlap 1 } \\ && \text{is an initial object in $\leftcat{ 1 \downarrow {\rightadj R} }$} & \leftcat\Set \rlap{ \kern.5em \rightadj{ \xleftarrow[\textstyle R]{\kern12em} } } &&&& \rightcat\calR & \leftcat\calL \rlap{ \kern.5em \leftadj{ \xrightarrow[\textstyle L]{\kern12em} } } &&&& \rightcat{\Set\op} & \leftcat{\calL\op} \rlap{ \kern.5em \leftadj{ \xrightarrow[\textstyle L]{\kern10em} } } &&&& \rightcat{\Set} \\ \hline &&&&& \leftcat \calI &&&&& \leftcat \calI &&&&& \leftcat \calI \\ 4. & \text{left lift} & \leftadj{ \lambda_0 : {\leftcat 1} \Rightarrow r_0 \rightadj R } && \leftcat{ \llap 1 \swarrow } & \smash{ \lower1ex{ \leftadj{ \stackrel {\lambda_0 = \eta} \Longrightarrow } } } & \leftadj{ \searrow \rlap{r_0 = \leftcat 1 L} } &&& \rightadj{ \llap{ \rightcat 1 R = l_0 } \swarrow } & \smash{ \lower1ex{ \rightadj{ \stackrel {\rho_0 = \epsilon} \Longrightarrow } } } & \rightcat{ \searrow \rlap 1 } &&& \rightadj{ \llap{ \rightcat 1 R = l_0 } \swarrow } & \smash{ \lower1ex{ \rightadj{ \stackrel {\rho_0} \Longleftarrow } } } & \rightcat{ \searrow \rlap 1 } \\ && \text{is a left lift of $\leftcat 1$ through $\rightadj R$ } & \leftcat\Set \rlap{ \kern.5em \rightadj{ \xleftarrow[\textstyle R]{\kern12em} } } &&&& \rightcat\calR & \leftcat\calL \rlap{ \kern.5em \leftadj{ \xrightarrow[\textstyle L]{\kern12em} } } &&&& \rightcat{\Set\op} & \leftcat{\calL\op} \rlap{ \kern.5em \leftadj{ \xrightarrow[\textstyle L]{\kern10em} } } &&&& \rightcat{\Set} \\ \hline &&& {} \rlap{ \kern3em \leftadj{ \text{left adjoint} } } &&&&& {} \rlap{ \kern3em \rightadj{ \text{right adjoint} } } \\ \hline &&& \leftcat{ \calI \rlap{ \xleftarrow [\kern11em] {!} } } &&&& \leftcat{l \downarrow {\rightadj R} } & \rightcat{ { {\leftadj L} \downarrow r } \rlap{ \xrightarrow [\kern11em] {!} } } &&&& \rightcat{\calI} \\ 5. & \text{left adjoint} & \leftadj{ (r_0 = \leftcat l L) \langle \lambda_0\rangle } & \leftcat{ \llap l \big\downarrow } && \smash{ \leftcat{ \stackrel \lambda \Longrightarrow } } && \rightcat{\big\downarrow \rlap r } & \leftcat{ \llap l \big\downarrow } && \smash{ \rightcat{ \stackrel \rho \Longrightarrow } } && \rightcat{\big\downarrow \rlap r } \\ && \text{is an initial object in $\leftcat{ l \downarrow {\rightadj R} }$} & \leftcat\calL \rlap{ \kern.5em \rightadj{ \xleftarrow[\textstyle R]{\kern12em} } } &&&& \rightcat\calR & \leftcat\calL \rlap{ \kern.5em \leftadj{ \xrightarrow[\textstyle L]{\kern13em} } } &&&& \rightcat\calR \\ \hline &&&&& \leftcat \calI &&&&& \rightcat \calI \\ 6. & \text{left lift} & \leftadj{ \lambda_0 : {\leftcat l} \Rightarrow r_0 \rightadj R } && \leftcat{ \llap l \swarrow } & \smash{ \lower1ex{ \leftadj{ \stackrel {\lambda_0 = \eta} \Longrightarrow } } } & \leftadj{ \searrow \rlap{r_0 = \leftcat l L} } &&& \rightadj{ \llap {\rightcat r R = l_0} \swarrow } & \smash{ \lower1ex{ \rightadj{ \stackrel {\rho_0 = \epsilon} \Longrightarrow } } } & \rightcat{\searrow \rlap r} \\ && \text{is a left lift of $\leftcat l$ through $\rightadj R$ } & \leftcat\calL \rlap{ \kern.5em \rightadj{ \xleftarrow[\textstyle R]{\kern12em} } } &&&& \rightcat\calR & \leftcat\calL \rlap{ \kern.5em \leftadj{ \xrightarrow[\textstyle L]{\kern13em} } } &&&& \rightcat\calR \\ \hline &&& \leftcat{ \hom l \calL {\rightcat r \rightadj R} } \rlap{ \kern.5em \leftcat{ \xleftarrow [ \leftadj{ \widehat{\lambda_0 = \eta} } ] {\kern10em} } } &&&& \rightcat{ \hom { \leftadj{ (r_0 = \leftcat l L) } } \calR r } & \leftcat{ \hom l \calL { \rightadj { (\rightcat r R = l_0) } } } \rlap{ \kern.5em \rightcat{ \xrightarrow [ \rightadj{ \widehat{\rho_0 = \epsilon} } ] {\kern10em} } } &&&& \rightcat{ \hom {\leftcat l \leftadj L} \calR r } \\ &&& {} \rlap{ \kern-2em { \leftadj{(\lambda_0 = \eta)} } \cdot {\rightadj{ (\rightcat\rho R) } } } && \leftcat{\leftarrow\kern-.2em \shortmid} && \rightcat\rho & \leftcat\lambda && \rightcat\mapsto && {} \rlap{ \kern-4em \leftadj{(\leftcat\lambda L)} \cdot \rightadj{(\rho_0 = \epsilon)} } \\ \end{array} } \]
