(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not as dreadful as they might appear?" but it was too long.)
This is a pretty basic question which may be more appropriate for another website. However, the examples I am looking for should appeal to a working mathematician. That is why I am asking here.
Tomorrow I shall have lunch with a mathematician whom notions of limits and colimits make nervous in general. He feels that calculating small examples may help him overcome his fear. More precisely, what he is asking for are examples of various small diagrams (including fancy ones, for instance with loops) in familiar categories (topological spaces, abelian groups, &c.) whose limit or colimit we could calculate together over lunch, in the hope that he would get a better understanding of what are limits and colimits when they are taken over diagrams other than those giving pullbacks or pushouts (of which he already has a feeling).
Do you know of some particular instances of diagrams, in categories familiar to the working mathematician, whose calculation of the limit or colimit seems particularly illuminating? Or at least which could help a nervous mathematician overcome his fear of general limits and colimits?
I could come up with ad hoc examples, but perhaps there is better than that?
EDIT: Sorry for the tardy edit. I went to sleep after asking the question and just woke up thinking "I should have made it Community Wiki, added a big-list tag and provided more details". Thanks for the answers so far. (By the way, I myself am unsure as to what extent this question is appropriate for MO, but if the three votes to close could be shortly explained I would nevertheless appreciate it.)
The examples should appeal to a mathematician working in geometry and topology. For some reasons he really would like to make concrete computations. It seems that he has been faced with the following situation (to which I myself have never been faced; that is why I am asking here, in the hope that someone else already has been in the same situation as his): he is given a fancy (not the most usual) small diagram in Top or Ab or whatnot and wants to compute the limit or the colimit. Somehow he is afraid of this. I feel that my question is somewhat too broad and unprecise, but this is the best I have come up with given what I was asked myself.
Best Answer
I'm not sure what would work for this individual, but I'd be tempted to turn this around, Jeopardy! style. That is, instead of being presented with a diagram and trying to compute its limit/colimit, take some construction and devise a diagram which naturally expresses the construction as a limit or colimit.
So for example, this might be too easy, but consider the construction $X/A$ where $A$ is a subspace of a topological space $X$. Is this naturally a limit or colimit? Well, it's a colimit, but of what? Again, this may be too easy since your friend is comfortable around pushouts. For extra credit: what is the sensible meaning of $X/\emptyset$?
Or, take the graph of a function like $y = x^2$. Can this be thought of as a limit or colimit? This time it's a limit, namely the equal-izer of two functions from $\mathbb{R}^2$ to $\mathbb{R}$. (There's a more general lesson to be learned here, that limits are generally loci of suitable equations.)
How about the localization $\mathbb{Z}[1/p]$ where we invert a prime? Perhaps a little harder, do the same for the localization $\mathbb{Z}_p$. Or (would this be too familiar?) how would you express the $p$-adics as a limit?
Or, come up with the condition that a presheaf over a space is a sheaf. This might be either too familiar or too abstract, however. It might be best to take more concrete examples like the ones above. These are all off the top of my head, though, and somewhat untested by me personally.