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

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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

$$\mathrm{colim}_{(C,f)} F (C) \ .$$

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Oh thanks I'll take a deeper look to understand the Kan extension right now. Sorry if I misphrased the first question, I was actually asking the other way around, can all direct (co)limits be seen as universal (co)cone ? –  Bogdan Oct 4 '11 at 12:22
Actually i confused a bit the notations. To make senes, the initial functor is $\Phi \colon \mathcal{C} \to \mathcal{D}$ and not $F$. I understand it now with your easier notation $\mathrm{colim}_{(\mathbf{\Phi} \downarrow D)} ( F\circ U)$. But what does it mean when there is an object instead of a functor next to the limit ? Is it interpreted the same, writing the functor $F$ or the object $F(C)$ ? Many thanks ! –  Bogdan Oct 4 '11 at 12:52
I'm adding some more hints on my answer. –  a.r. Oct 4 '11 at 12:54