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