This section now relate 3 concepts about universality explained previously and obtain some fundamental results.
In the previous section, it is mentioned that the limit is the universal cone. We’ll rephrase it in terms of other universality conditions, starting from the equivalent but familiar definition of cones.
A cone with the vertex $A\in \mathcal{A}$ and the diagram $D\in[\textbf{I},\mathcal{A}]$ consists of $|\textbf{I}|$ number of morphisms starting from $A$ to each $D(I)$, with a commuting condition. We can notice some similarity of definition with that of natural transformation; and indeed is a natural transformation! The codomain functor of the natural transformation is $D$ for sure, and if the domain functor send all $I$s to $A$ and morphisms to $1_A$, we have a equivalent definition for the cone. More precisely, a cone on a diagram $D$ with vertex $A$ is just the morphism from $\Delta_\textbf{I} A$ to $D$, where $\Delta_\textbf{I} A$ is defined by the following:
$$ \Delta_\textbf{I} A : \textbf{I} \rightarrow \mathcal{A} : I\mapsto A,\,f\mapsto 1_A $$
The symbol $\Delta_\textbf{I}$ is used emphasizing the single object $A$ is repeated $|\textbf{I}|$ times, which is analogous to diagonalization. Indeed, we denote $\Delta_\textbf{I}$ as a diagonal functor, making $|\textbf{I}|$ copies of each object and morphism in $\mathcal{A}$.
$$ \Delta_\textbf{I}:\mathcal{A} \rightarrow [\textbf{I},\mathcal{A} ]: A \mapsto \Delta_\textbf{I} A:g\mapsto \left\{g\right\}_{I\in \textbf{I}} $$
In short, we have a notation of a set of cones by $\text{Cone}(A,D):=[\textbf{I},\mathcal{A}](\Delta_\textbf{I} A,D)$, or equivalently, a set of natural transformations.
Fixing the diagram $D$, the limit of it should belongs to $\text{Cone}(L,D)$ for some $L\in \mathcal{A}$, and is a universal element among $\text{Cone}(-,D)$. Thus, regarding $\text{Cone}(-,D)$ as a functor $H_D \circ \Delta_\textbf{I}^\text{op} :\mathcal{A}^\text{op} \rightarrow \textbf{Set}$(which is presheaf), $D$ has a limit if and only if the functor has a universal element, which is equivalent to the representability of the functor.
Proposition. For a diagram $D\in[\textbf{I},\mathcal{A}]$,
$$ \exist \lim_{\leftarrow\textbf{I}} D\Longleftrightarrow \text{Cone} (-,D)= H_D\circ \Delta_\textbf{I}^\text{op}=H_{\exist L} \in \hat{\mathcal{A}}\ , $$
and $\displaystyle\lim_{\leftarrow\textbf{I}}D ,\, L$ coincides(up to isomorphism).
Along with varying the diagram $D$, we now assume $\mathcal{A}$ has a limit of shape $\textbf{I}$. This allows making a well-defined functor $\displaystyle \lim_{\leftarrow\textbf{I}}:[\textbf{I},\mathcal{A}]\rightarrow \mathcal{A}$: Not only it assigns each diagram to its limit, it also assigns each natural transformation between diagrams to unique morphism between limits, using the limit’s property.
Indeed, using this functor, the limit-ness could be stated in even shorter term: The right adjoint of the $\Delta_\textbf{I}$.
Proposition. If $\mathcal{A}$ has a limit of shape $\textbf{I}$, then $\displaystyle \lim_{\leftarrow \textbf{I}}$ is a functor and $\displaystyle \Delta_\textbf{I}\dashv\lim_{\leftarrow \textbf{I}}.$