I am reading Category Theory for Programmers and I am having some trouble in Part 2, Chapter 2: Limits and Colimits.
The author writes:
Now that we have two functors, we can talk about natural transformations between them. So without further ado, here’s the conclusion: A functor $D$ from $I$ to $C$ has a limit $\lim D$ if and only if there is a natural isomorphism between the two functors I have just defined: $$C(c, \lim D) ≃ Nat(Δc, D)$$
How can you express the existence of this natural isomophism as a commutative diagram?
The formula you write above is actually a description of the relevant diagram!
For an object $c\in \mathcal C$, and $D\in\mathcal{C^D}$, what is a natural transformation $\gamma\in\mathrm{Nat}(\Delta c,D)$ like? Like usual, it's going to need to satisfy the commutativity condition $\gamma_{d'}\circ\Delta c(f)=D(f)\circ\gamma_d$ for all $f:d\to d'$ in $\mathcal D$; however, no matter what $f$ is, by definition $\Delta c(f)$ is just $id_c$, so we can simplify that condition to $\gamma_{d'}=D(f)\circ \gamma_d$. But that makes $\gamma$ precisely a cone over $D$. In particular, the limiting cone is just the cone $\alpha:\Delta\circ\lim D\to D$ corresponding to $id_{\lim D}$ by the above isomorphism.
Moreover, given such a $\gamma$, the isomorphism above means it corresponds to a unique morphism $\mathcal C(c,\lim D)$ that is the arrow $f_\gamma$ typically drawn as "the dotted line" on paper. To see why this has the desired property, remember that the isomorphism $\phi:\mathcal C(-,lim D)\to\mathrm{Nat}(\Delta -,D)$ is natural, so that $$\mathrm{Nat}(\Delta(f_\gamma),D)\circ\phi_{\lim D}=\phi_c\circ\mathcal C(f_\gamma,\lim D).$$ If you chase what happens to $id_{\lim D}$ in this square, you'll see that this says exactly that composing the limiting cone with $f_\gamma$ gives you the cone $\gamma$.