Here's another theorem from Leinster's book (p. 159) where I got stuck:
Just as in my previous question, I don't see how this sequence of isomorphisms establishes the claimed result. To prove the theorem, one has either use the definition of limit preservation or use a result quoted here, and I don't see how any of these is reflected in the proof. I tried to track what those isomorphisms do explicitly but I got stuck:
$$f\mapsto \epsilon_{\lim D}\circ F(f)\mapsto (p_i\circ (\epsilon_{\lim D}\circ F(f)))_{i\in I}\mapsto ??$$
(I know how the isomorphism $\mathscr B(F(A),D)\to \mathscr A(A,G\circ D)$ works, but once we add $\lim$ on the left-hand sides of these sets, things become unclear, and I don't know where $(p_i\circ (\epsilon_{\lim D}\circ F(f)))_{i\in I}\in\lim \mathscr B(F(A),D)$ should be sent to.)
Do I even need to do this? Or what's the easiest way to understand the proof?
In the proof, Leinster is using the usual notion of limit preservation. That is, he's trying to show that $G(\mathrm{lim}D)\cong \mathrm{lim}(G\circ D)$. His strategy to do this is to show that $G(\mathrm{lim}D)$ represents the functor $\mathrm{Cone}(-,G\circ D)$ ($\mathrm{Cone}(-,G\circ D)$ is the functor that sends an object $A\in\mathscr{A}$ to the category of cones over $G\circ D$ with apex $A$). One can in fact define limits as objects that represent the cone functor. This all amounts to showing that $\mathrm{Cone}(-,G\circ D)\cong\mathscr{A}(-,G(\mathrm{lim}D))$, but this is exactly what the chain of isomorphisms provides!
The isomorphism witnessing $\mathrm{lim}\mathscr{B}(F(A),D)\cong\mathrm{lim}\mathscr{A}(A,G(D))$ where you are stuck can be obtained by adopting the perspective that the limit is functorial. That is, for a category $\mathscr{C}$ admitting limits of shape $I$, there is a functor $$\mathrm{lim}_{I}:\mathscr{C}^I\longrightarrow \mathscr{C}$$ $$D\mapsto \mathrm{lim}_{I}D$$ Now, the adjunction provides us a natural isomorphism of functors: $$\mathscr{B}(F(A),D(-))\cong\mathscr{A}(A,G(D(-)))\in\mathrm{Set}^{I}$$ (assuming local smallness for the simplicity of this argument). Since functors preserve isomorphisms, we immediately recover the desired isomorphism: $\mathrm{lim}\mathscr{B}(F(A),D)\cong\mathrm{lim}\mathscr{A}(A,G\circ D)$. Unfolding the definitions here should provide you a way to continue tracing the chain of isomorphisms.