I'm trying to work out this example in detail:
The aim is to verify this result (the first isomoprphism in the picture) in the above special case:
The relevant isomorphism is given by:
First we need to compute $\lim_I D^\bullet$. By definition, it is an object $\lim_I D^\bullet$ of $[J,\mathscr C]$ together with projections $(p_i:\lim_I D^\bullet\to D^\bullet(i))_{i\in\{\bullet_1,\bullet_2\}}$ such that for any object $F$ in $[J,\mathscr C]$ arrows $\alpha_1:F\to D^\bullet(\bullet_1)$ and $\alpha_2:F\to D^\bullet(\bullet_2)$ there is a unique $\alpha:F\to \lim_I D^\bullet$ such that $p_i\circ\alpha=\alpha_i$ for $i=1,2$. It's the product $f_1\times f_2$ in the category $[J,\mathscr C]$. Is there a more explicit description? For example, can one explicitly define this limit (which is an element of $[J,\mathscr C]$) by saying which objects of $\mathscr C$ the objects $\bullet_1,\bullet_2$ map to? If so, how to understand where they should be mapped? And how to understand how $p_i$ should be defined?
Now we need to compute $\lim_J f_1\times f_2$. I don't see how to do this. According to the first display of Example 6.2.9, the result should be, apparently, $(S_{11}\times S_{21})\times (S_{12}\times S_{22})$, but how to obtain this?
Edit: I found the answers to the above here.
As for $\lim_{I\times J} D$, is there nothing to do here? By definition, this is a 4-product $\prod_{i,j} D(\bullet_i,\bullet_j)$. But still one question about this: $\prod_{i,j} S_{ij}$ is just a notation for a specific object of $\mathscr C$, and I can't write it as $S_{11}\times S_{12}\times S_{21}\times S_{22}$, can I?
The first display of Example 6.2.9 shows 2 isomorphisms. Does it not follow that those objects are isomorphic to e.g. $(S_{21}\times S_{12})\times (S_{11}\times S_{22})$?
Finally, how does the "more generally" part follow? And why is it "more generally"? Should we consider different $I, J$ and apply the proposition again? If so, which $I,J$?