Here's something that I'd like to use in my thesis.. but Im feeling too lazy to write a proof of it, I feel pretty sure this is correct though. I have a feeling that this can be found in a book on category theory. So maybe someone can point me to a reference (I have only used Adamék, Herrlich and Strecker so far).


Conjecture:
Short Statement: pullback of inverse limits is the inverse limit of pullbacks 


Long Statment:
Let $I$ be a directed set and let $\mathbf D,\mathbf E: I \rightarrow \mathcal C$ be two diagrams to a complete category. Let $C$ be an object in $\mathcal C$ and suppose that we have natural sinks $\mathbf Di \rightarrow C$ and $\mathbf Ei\rightarrow C$ (Mac Lane calls these "cones"). Let $A$ and $B$ be the inverse limit of $\mathbf D$ and $\mathbf E$ respectively. We get another diagram that goes to the pullback namely $\mathbf D \times\_C \mathbf E : I \rightarrow \mathcal C \times\_C \mathcal C$. The claim is that the inverse limit of $\mathbf D \times\_C \mathbf E$ is actually the fiber product (or pullback, however you want to call it) $A \times\_C B$ (where $A\rightarrow C$ and $B\rightarrow C$ are canonical map that results from the inverse limit and the natural sinks).



Need a reference for cones and limits that does this...
Yes -- filtered/directed colimits commute with finite limits. See Mac Lane, *Categories for the Working Mathematician*, theorem IX.2.1.


Edit: Oh, I thought you meant colimits, but it seems you meant limits. But limits always commute with each other (CWM IX.2).


