For $C$ a locally small category, $J$ an essentially small category and $F\colon J\rightarrow C$ a functor, the limit of $F$, if it exists, can be defined as a representation of the functor $\operatorname{Cone}(-,F) \colon C^{\mathrm{op}} \rightarrow \mathsf{Set}$. This functor sends an object in $C$ to its set of cones. It can more succinctly/formally be defined as the composition of functors $$C^{\mathrm{op}}\xrightarrow{\Delta^{\mathrm{op}}}[J,C]^{\mathrm{op}}\xrightarrow{\operatorname{Hom}_{[J,C]}(-,F)}\mathsf{Set}.$$ By $\Delta: C\rightarrow [J,C]$ we denote the diagonal functor that sends an object $c\in C$ to the composite $\Delta(c): J\rightarrow \ast \rightarrow C$, where $\ast \rightarrow C$ maps the unique object in the terminal category $\ast$ to $c$. A morphism $f:c\rightarrow d$ is sent to the natural transformation $\Delta(c)\Rightarrow \Delta(d)$, which has $f$ as its components if $J$ is inhabited and is empty otherwise.
On the nLab a limit of $F$ is defined slightly differently. If I understand correctly, there, a limit of $F$ is a representation of the functor $$C^{\mathrm{op}}\xrightarrow{G}[J,\mathsf{Set}]\xrightarrow{H}\mathsf{Set},$$ where $G(c)=\operatorname{Hom}_C(c,F-)$ and $H(L)=\operatorname{Hom}_{[J,\mathsf{Set}]}(\operatorname{pt},L)$. Here $\operatorname{pt} \colon J \rightarrow \mathsf{Set}$ is the functor that sends every object in $J$ to the singleton set.
How are these two definitions related? Is there a quick way to see that they are equivalent?
$\newcommand{\op}{^{\mathsf{op}}}\newcommand{\C}{\mathsf{C}}\newcommand{\set}{\mathsf{Set}}$The nLab definition given there is a restatement of point $(6)$ in my answer here. I mentioned it was a close cousin to the cone definition, here's why:
You want to compare representing objects $L$ for the two functors $\C\op\to\set$:
So it remains to demonstrate an isomorphism between these functors. A natural transformation $\Delta(c)\to F$ is just a cone over $F$, a family of arrows $c\to F(j)$ which satisfy some coherence conditions. A natural transformation $\ast\implies\C(c,F(-))$ is a family of functions $\ast\to\C(c,F(j))$ which is just a family of elements of $\C(c,F(j))$, $j\in J$, that satisfy coherence conditions. Recall functions $1\to X$ are in correspondence with elements of $X$, and this observation gives rise to concepts of generalised element. An element of $\C(c,F(j))$ is just an arrow $c\to F(j)$; the coherence conditions are exactly those required to have a cone, a natural transformation $\Delta(c)\implies F$.
In symbols, for some natural $\phi:\ast\implies\C(c,F(-))$ we define the arrows $\lambda_j:=\phi_j(\ast)\in\C(c,F(j))$ and these form the components of the natural transformation $\Delta(c)\implies F$. It's easy to see this process is natural in $c\in\C\op$.
Lastly, isomorphic functors entail isomorphic representing objects, so these two notions of limit agree up to isomorphism.
A minor digression.
This equivalence is a special case of a tensoring (assuming $\C$ admits all copowers): $$\C^J(W\otimes c,F)\cong\set^J(W,\C(c,F))$$Where for a functor $W:J\to\set$ we make a functor $W\otimes c$ by mapping $j\in J$ to $\bigsqcup_{W(j)}c$ (i.e. applying the usual tensoring at the a pointwise level). In the special case of the trivial weight $W\equiv\ast$, or $W\equiv1$ if you prefer, $\bigsqcup_{W(j)}c\cong c$ and this functor $W\otimes c$ is isomorphic to $\Delta(c)$. I'm pretty sure that if we assume $\C$ is complete then there will be a cotensoring $\set^J(W,\C(c,F))\cong\C(c,W\pitchfork F)$; in the special case of $W\equiv\ast$, $W\pitchfork F$ is just a limit for $F$, as expected by the representable definition.