Can we compare limits over different diagrams? Fix an index (small) category $I$. Let's say our category $\mathcal{C}$ has limits of type $I$. In this case, $\varprojlim_I:\mathcal{C}^I \to \mathcal{C}$ is a functor.
Let's say our category $\mathcal{C}$ also has limits of type $J$. In this case, $\varprojlim_J: \mathcal{C}^J\to \mathcal{C}$ is also functor.
How do these relate?
Let me clarify. Suppose now our category $\mathcal{C}$ is complete, i.e. has all (small) limits. Can we somehow define a functor $\varprojlim$ which takes any diagram of any type and outputs its limit?
Motivation: let $R$ be a ring (commutative for simplification). It's easy to formalize the statement that the isomorphism $A\otimes_R \bigoplus_{i\in I} B_i \simeq \bigoplus_{i\in I} (A\otimes_R B_i)$ of $R$-modules is "natural in $A$".
But can it be also natural in $\{B_i\}_{i\in I}$? Yes:
This isomorphism is a particular case of the more general statement $A\otimes_R \varinjlim_I F\simeq \varinjlim_I A\otimes_R F$, where $F:I\to R-\mathrm{Mod}$ is a functor. This isomorphism is natural also in the second variable, meaning that there is a natural isomorphism between two appropiate functors.
But what if we also let $I$ vary?
Going back to the particular case of the direct sum, there is this proposition (taken out of Rotman, Introduction to Homological Algebra, page 87):

Now, this "naturality" involves also varying the index set $I$. How can this "naturality" really be expressed as a natural isomorphism between functors? Can it be generalized for arbitrary colimits?
 A: Maybe what you're looking for is the following: a functor $\phi : I \longrightarrow J$ is called final if, for every object $j\in J$, the comma category $(j\downarrow \phi)$ is non-empty and connected. That means that, for every object $j$, there is an object $i\in I$ and an arrow $j \longrightarrow \phi i$ and two any such arrows can be connected with a finite commutative diagram of the form
$$
\begin{matrix}
j          & = & j          & = & j          & \dots & j          & = & j \\
\downarrow &   & \downarrow &   & \downarrow & \dots & \downarrow &   & \downarrow  \\
\phi i     & \leftarrow & \cdot & \rightarrow & \cdot & \dots & \cdot     & \rightarrow & \phi i' 
\end{matrix}
$$
Let $F : J \longrightarrow \mathcal{C}$ be any functor and $\mu : F \longrightarrow \mathrm{colim}_j Fj$ and $\mu' : I \longrightarrow \mathrm{colim}_i F\phi i$ be the colimiting cones. You always have a canonical morphism
$$
h : \mathrm{colim}_i F\phi i \longrightarrow \mathrm{colim}_j Fj
$$
such that $h_i\mu'_i = \mu_{\phi i}$ for every $i \in I$.
Now, assume that $\phi$ is final and that $\mathrm{colim}_i F\phi i$ exists. Then $\mathrm{colim}_jFj$ also exists and $h$ is an isomorphism.
Dualising, you get cofinal (initial) functors and an analogous result for limits. You can find all this stuff in Mac Lane's "Categories for the working mathematician".
A: There is a notion of Kan extension generalizing the notion of (co)limit.
Namely, suppose the category $C$ has enough limits. Then for any functor $\phi\colon I\to J$ there is a functor $\operatorname{Ran}_\phi\colon C^I\to C^J$ (defined by a kind of "fiberwise limit" construction, if you will). And for $\phi\colon I\to pt$, $\operatorname{Ran}_\phi$ is just $\lim_I$.
Finally, Lan is functorial in the obvious sense (because $\operatorname{Ran}_\phi$ is the adjoint functor to $\phi^*$). In particular, $\lim_I=\lim_J\circ\operatorname{Ran}_\phi$.
Tagline: functorial limit = Kan extension.
