Goal: I want to phrase "continuity of a functor $F$" as an isomorphism (between functors): $\lim\circ F_*\cong F\circ\lim$.
Let $\mathcal C,\mathcal D,I$ be categories, and let $F:\mathcal C\to\mathcal D$ be a functor. Suppose that all limits of shape $I$ exist in both $\mathcal C$ and $\mathcal D$; we can therefore consider the functor $\lim$ which sends a diagram of $I$ to its corresponding limit. I denote by $[\mathcal A,\mathcal B]$ the functor category consisting of functors from $\mathcal A$ to $\mathcal B$.
Consider the composite functors: $$ \lim\circ F_*:[I,\mathcal C]\to [I,\mathcal D]\to \mathcal D $$ and $$ F\circ\lim:[I,\mathcal C]\to \mathcal C \to\mathcal D $$ One can show that if $F$ preserves limits of shape $I$, then the two functors above are isomorphic. I'm wondering if it also goes the other way around? So basically the question becomes, if we have for any diagram $D$ an isomorphism $F\lim_i D_i\cong\lim_i (FD_i)$ such that for any natural transformation $\nu:D\to D'$ we have a commutative diagram
is it then true that
commutes as well, where $p_i$ is the projection map corresponding to $\lim D$ and $\pi_i$ is the projection map corresponding to $\lim(FD)$?
Note that the isomorphism $F(\lim_i Di)\cong\lim_i FDi$ clearly says that $F$ sends the vertex of the limit of $D$ to the vertex of the limit of $FD$, but for the definition of preservation of limits I also need that the universal cone $p:\Delta \lim D_i\to D$ is mapped by $F$ to a universal cone, i.e. that $Fp:\Delta F\lim D_i\to FD$ is a universal cone. It is this part that I'm trying to prove (if it's possible) assuming that we have an isomorphism $F\circ\lim\cong \lim\circ F$ of functors.