What type of limit is used when? I have some trouble understanding what type of limit is sometimes used. The definition I have is the one when a limit is defined as a universal cone over a functor (say from a small category). 
So my first question is : Can we always reduce a limit to this type, because sometimes I can't identify the underlying small category and functor ? 
I think I understand the limit of some directed set (for example the Stalk of a sheaf at a point), can this be reduced to one of the above type, but we just don't do it for convenience, or are they different notions ?
My question was motivated by the construction of a left Kan extension. Given a functor $\Phi \colon \mathcal{C} \to \mathcal{D}$, and a cocomplete category $\mathcal{E}$, there is a precomposition functor $\Phi^{\ast} \colon \mathcal{E}^{\mathcal{D}} \to \mathcal{E}^{\mathcal{C}}$ and we want to find him a left adjoint $\Phi_{\ast}$. 
It seems like an adjoint functor can be defined for all objects (a functor) $F \in \mathcal{E}^{\mathcal{C}}$ by $\Phi_{\ast}F \colon \mathcal{D} \to \mathcal{E}$ that sends $D \mapsto \text{colim}_{\textbf{Simp}_{\Phi}(D)} F(C)$, where $\textbf{Simp}_{\Phi}(D)$ is the category with objects being $\{ (C,f) \ | \ C \in \mathcal{C}, \ \Phi(C) \stackrel{f}{\to} D \}$ (a comma category ?). The thing is that i don't get the meaning of this colimit, since it is indexed by a category and $F(C)$ is an object in the required category... Can someone help me clarify these notions ?
Thanks yo
 A: I'm not sure I understand your doubts, but let's try to say something.
Yes. All (co)limits are defined as a universal (co)cone over a functor from a (small, "indexing") category.
No. You cannot reduce (co)limits to direct (or filtered) (co)limits. For instance, in the category of abelian groups, the direct sum $A \oplus B$ is a colimit, but not a filtered one. It's the colimit of the functor $F : I \longrightarrow \cal{A}$, where $\cal{A}$ is the category of abelian groups, and the indexing category $I = \left\{ 0,1 \right\}$ is the category with just two objects and just two morphisms (the identities) and $F$ the functor $F(0) = A$ and $F(1)=B$. Though a very simple one, $I$ is not a filtered category and you can write, if you wish,
$$
A \oplus B = \mathrm{colim}_I F
$$
As for the example of the Kan extension, the indexing category is now the comma category $(\mathbf{\Phi} \downarrow D) = \mathbf{Simph}_{\mathbf{\Phi}} (D)$. But $F(C)$ is an object of $\cal{D}$: so, where is the problem? What is your "refered category"?
Maybe you would understand better the colimit  written with more detail, using the forgetful functor $U :(\mathbf{\Phi} \downarrow D) \longrightarrow \cal{C} $, which sends every obect $(C, f) \in (\mathbf{\Phi} \downarrow D)$ to $C$, as
$$
\mathrm{colim}_{(\mathbf{\Phi} \downarrow D)} ( F\circ U ) \quad \text{?}
$$
EDIT. You could also write this same colimit as
$$
\mathrm{colim}_{(C,f)}  F (C)  \ .
$$
