Certain preserved limits give left adjoints

WORK IN PROGRESS!

Here we use the fact that (the cone  $\boxed{    \pi : \leftcat{1_{(l \downarrow \rightadj R)} }\lim \Rightarrow \leftcat{1_{(l \downarrow \rightadj R)} }: \leftcat{(l \downarrow \rightadj R)} \to \leftcat{(l \downarrow \rightadj R)}    }$) is (a natural transformation). 

Hence (the following triangles) commute for all ($\rightcat r \leftcat{[\lambda]}\in \leftcat{(l \downarrow \rightadj R)}$) and ($\rightcat f$ as shown):

\[  \boxed{  \begin{array}  {c|cccccc|cccccc}  \leftcat{1_{(l \downarrow \rightadj R)} } & \kern0em   &  \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)} &  {} \rlap{   \kern-1em \xrightarrow[\smash{\kern15em}]{\textstyle \rlap{\pi_{\rightcat r[\lambda]}} }  }   &&&  \rightcat r \leftcat{[\lambda]} & \kern0em   &  \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)} &  {} \rlap{   \rightcat{  \kern-1em \xrightarrow[\smash{\kern15em}]{\textstyle f }  }  }   &&&  \rightcat r  \leftcat{[\lambda]}     \\      \llap{\pi} \Big\Uparrow &  &&  \llap{ \pi_{\rightcat{ (\black Q\lim_\calR) } \leftcat{ \big[\overline{\lambda}\big] }}  } \nwarrow  &  \pi_{\pi_{\rightcat r \leftcat{[\lambda]}}}  &  \nearrow \rlap{\pi_{\rightcat r[\lambda]}} &&  &&  \llap{ \pi_{\rightcat{ (\black Q\lim_\calR) } \leftcat{ \big[\overline{\lambda}\big] }}  } \nwarrow  &  \pi_{\rightcat f}  &  \nearrow \rlap{ \pi_{\rightcat r[\lambda]} }  \\    \leftcat{   1_{(l \downarrow \rightadj R)} \lim_{(l \downarrow \rightadj R)  }   }  & & &&  \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)}  & & && &&    \leftcat{  1_{(l \downarrow \rightadj R)} \lim_{(l \downarrow \rightadj R)}  }    \\ \end{array}  }  \]

To reiterate, (the two triangles above) commute for all ($\rightcat r \leftcat{[\lambda]}\in \leftcat{(l \downarrow \rightadj R)}$) and ($\rightcat f$ as shown) because ($\rightcat\pi$ is a cone).

Using (naturality of $\pi$ at $\pi_{\rightcat r \leftcat{[\lambda]}}$), i.e. $ \pi_{\pi_{\rightcat r \leftcat{[\lambda]}}}$,  plus 

(the uniqueness clause in the universal property of $\leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)}$), 

we get that $\pi_{\rightcat{ (\black Q\lim_\calR) } \leftcat{ \big[\overline{\lambda}\big] }} = 1_{  \leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)}  }$.

This, plus (naturality of $\pi$ at $\rightcat f$), i.e. $\pi_{\rightcat f}$,

gives $\rightcat f = \pi_{\rightcat r[\lambda]}$.

This in turn gives us that 

$\leftcat{1_{(l \downarrow \rightadj R)} }\lim_{(l \downarrow \rightadj R)}$ is 

(an initial object in ${(l \downarrow \rightadj R)}$},

which is equivalent to

$\rightcat{ (Q\lim_\calR) } \leftcat{ \big[\overline{\lambda}\big] }$ being 

(a universal arrow from $\leftcat l$ to $\rightadj\calR$) 

which is equivalent to

$\rightcat{ (Q\lim_\calR) }$ being (a left adjoint $\leftcat  l \leftadj L$ for $\leftcat  l$) with ($\leftcat{ \overline{\lambda} }$ its unit).

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\[ \boxed{ \begin{array} {c|c|c} \kern2em & \kern2em & \kern2em \\ xx & yy & zz \\ \end{array} } \]

\[ \boxed{ \begin{array} {} && r \\ & \llap{\pi_r} \nearrow & \pi_{\pi_r} & \nwarrow \rlap{\pi_r} \\ \leftcat l\leftadj L & {} \rlap{ \kern-1em \xrightarrow[\pi_{\leftcat l\leftadj L}] {\kern8em} } &&& \leftcat l\leftadj L \\ \end{array} } \]