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?