On page 88 of Mac Lane’s CWM it is mentioned that adjoints of a diagonal functor with $J$ generated by $ \leftarrow \cdot \rightarrow $ are pushouts and pullbacks. I had no problems verifying this for pushouts. However, how does one get a pullback out of this $J$? Specifically, assuming we have an adjunction
$$ ( \Delta , \underleftarrow{\text{Lim}} , \varphi ): C \rightharpoonup C^{J} $$
Here any universal cone is given by a component of the counit, which can be taken as an image of identity under a certain component of $\varphi^{-1}$, but this diagram looks like
$\require{AMScd}$
$$\begin{CD}
\underleftarrow{\text{Lim}} @>>> \cdot\\
@VVV @AAA\\
\cdot @<<< \cdot
\end{CD}$$
which is clearly not a pullback square.
As such, my exact question is: Where in this construction is $J$ supposed to be “flipped” (if it is)?
Is this the $-^{\text{op}}$ functor’s doing in hom-set or something?
Best Answer
There is a typo in CWM. For getting pullbacks you should take $\cdot\rightarrow\cdot\leftarrow\cdot$, of course.
The duality between pullbacks and pushouts means that a pullback of a pair of morphisms $(f,g)$ in a category $C$ is a pushout of this pair in $C^{op}$. A pushout of $(f,g)$ in $C^{op}$ is a colimit of the functor $F_{(f,g)}\colon[\cdot\leftarrow\cdot\rightarrow\cdot]\to C^{op}$, hence it is a limit of the dual functor $(F_{(f,g)})^{op}\colon[\cdot\rightarrow\cdot\leftarrow\cdot]\to C$.
So you can get the adjunction for pullbacks in $C$ simply by taking the dual adjunction of the adjunction for pushouts in the dual category $C^{op}$. You start with the functor $\varinjlim\colon(C^{op})^J\to C^{op}$ and duality gives you the functor $\varprojlim\colon C^{(J^{op})}\to C$.