I am reading Fosco's [Coend Calculus][2] and Emily Riehl's [Categorical Homotopy Theory][3], Riehl's book motivates it in the following way,

Abstraction 1: Classical limits in terms of cones: Cones from an object $C$ to a system $F:I\to \mathcal{C}$ are thought of as natural transformations from  terminal functor, to the collection of natural transformations from the constant functor to the system. 
Instead of terminal functor, when one has a weight functor $W: I\to \mathbf{Sets}$, a natural transformation $\tau$ from $W$ to the functor $Hom_{\mathcal{C}}(C,F-)$ has as components set maps $\tau_i:Wi\to Hom_{\mathcal{C}}(C, Fi)$. So, each natural transformation gives a more complicated cone (there are different sets of connections from $C$ to $Fi$ for each $i$)


$$Hom_{\mathcal{C}}(C,\textstyle\lim^W F)\cong Hom_{\mathbf{Sets}^{I}}(W, Hom_{\mathcal{C}}(C, F-))$$
Each component natural transformation $\tau\in Hom_{\mathbf{Sets}^{I}}(W, Hom_{\mathcal{C}}(C, F-))$ is a set map $\tau_i:Wi\to Hom_{\mathcal{C}}(C, Fi)$, which assigns to elements of the set $Wi$ cones from $C$ to the system $F:I\to \mathcal{C}$. This is 'weighted' in the following sense,

Abstraction 2: I was thinking that $W:I\to \mathbf{Sets}$ assigns some 'generalised weightage' to the objects in the category $I$, weights measured by sets. By this I mean, when we assign weight in other branches of mathematics we are usually thinking of real numbers, and 'more weight' is based on the ordering of reals. Instead of this order relation we generalise 'more' to mean any set map. 


Intuition for Limits: The intuitive way I think of limits are as the object closest from a system, closest in terms of morphisms. By that I mean any other object which can be related to the system must be farther than the limit, that is, there must exist a map from limit to this object. How do I think of weighted limits? 

So, I feel weighted limit does the following: It is the object closest to the weight, in the sense that, the collection of maps $\{\lim^W F\to Fi\}$ is closest to the set $Wi$ (again in terms of morphisms) or something like this?


This all seems way too abstract for me, it felt like abstraction of things that are abstract generalizations of the intuitive things. 'Higher abstraction' of some sorts. But how does someone come up with such definitions, and why? and is my intuition correct? 


In the below diagram, the weighted limit $Lim^W F$ I guess should be such that the collection of maps from $Lim^W F$ to $Fj$ is closest to $Wj$, closest in the sense that given any other object $C$, there should exist a map from $Hom(C, Fj)$ to $Wj$
[![enter image description here][1]][1]

  [1]: https://i.sstatic.net/VP2kt.jpg
  [2]: https://arxiv.org/abs/1501.02503
  [3]: https://www.cambridge.org/core/books/categorical-homotopy-theory/556C7A200B521E61466BB7763C49DDA4