What does it mean for F(Lim X) to be 'naturally isomorphic' to Lim(F(X))

Question

Let F be a left adjoint functor. Given any diagram X, what does it mean for F(Lim X) to be naturally isomorphic to Lim(F(X)). What are the functors here which are naturally isomorphic?

Answer 1

The functors are $$\mathcal{C}^I\overset{F^I}{\longrightarrow}\mathcal{D}^I\overset{lim}{\longrightarrow}\mathcal{D}$$ and $$\mathcal{C}^I\overset{lim}{\longrightarrow}\mathcal{C}\overset{F}{\longrightarrow}\mathcal{D},$$ where $I$ is the domain of the functor $X$.

There is some possible abuse here, depending on whether or not $\mathcal{C},\mathcal{D}$ are actually (co)complete; an author could conceivably phrase this as if the limit functors exist when they don't really, as a kind of fiction to simplify the statement.

If they are not, then we can rephrase it, albeit in an uninteresting way. If $\bar{X}$ is the right Kan extension of $X$ along $!:I\to 1$, then we can say that $\overline{F\circ X}$ is naturally isomorphic to $F\circ\bar{X}$ as functors $1\to\mathcal{D}$. This is uninteresting because clearly we're just saying they're isomorphic as objects.

Personally, I don't think I've ever seen someone talk about natural isomorphism in the latter case, and I wouldn't recommend doing so.