Adjoint functors preserve limits/colimits 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?
 A: 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.
