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Let $F : J → C$ be a diagram of type $J$ in a category $C$. A cone to $F$ is an object $N$ of $C$ together with a family $ψ_X : N → F(X)$ of morphisms indexed by the objects $X$ of $J$, such that for every morphism $f : X → Y$ in $J$, we have $F(f) \centerdot ψ_X = ψ_Y$.

So, is $X$ here referring to a single object of $J$ or several objects of $J$ grouped together? (In terms of set, is $X$ an element of $J$ or a subset of $J$?)

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I think the question that you are asking pertains to identifying what $X$ is in the quoted portion. It is simply an object of the category $J$ and $f$ is merely a morphism in $hom_J(X,Y)$. – Karl Kronenfeld Mar 27 '13 at 12:59
Perhaps one or two answers are missing which explain the same thing once again? – Martin Brandenburg Mar 27 '13 at 17:20

$\hskip 0.1in$ categories

A diagram of type $J$ can be thought of as a copy of $J$ (formally, the image of the category $J$ under the functor $F$) sitting inside another category $C$. All of the objects and morphisms in $J$ are carried over to a relabelled diagram sitting inside $C$ (I omitted the arrows in purple and blue because I was lazy). The idea of a cone is that you can append an object $N$ to this diagram in the sense that you put the object in your sights and then attach a morphism $\psi_A:N\to FA$ for each $A\in{\rm Ob}(J)$, and as a result you will have a new commutative diagram (blue + light blue above).

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The cone is indexed by all objects of $J$.

Before dealing with limits in general, you should understand products and their universal property. Of course all factors are involved.

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There is a collection of $\psi_X$. One $\psi$ for each object in $J$. So if $J$ has 3 objects $a,b,c$ you have $\psi_a, \psi_b,\psi_c$. But, in general, you cannot choose the $\psi$ as you wish. The various $\psi$ have to respect some constraints among themselves: if there is a $f:X \to Y$ then, even if you can freely choose $\psi_X$, $\psi_Y$ is automatically determined by $\psi_Y = F(f) \centerdot \psi_X$.

So, to answer explicitly to the question in your title: each $X$ is a single object in $J$. The collection of $X$ is the collection of all objects in $J$

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