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I think that producing beautiful examples strongly depends on the background of your friend. Nevertheless, the following is the best I can find.

Let $\cal C$ be small; compute the following limits and colimits

1. $\varprojlim {\cal C}(x,-)$
2. $\varprojlim {\cal C}(-,x)$
3. $\varinjlim {\cal C}(x,-)$
4. $\varinjlim {\cal C}(-,x)$

where $\mathcal C(-,x)\colon {\cal C}^\text{op}\to \bf Set$, $\mathcal C(x,-)\colon {\cal C}\to \bf Set$ are representable hom-functor, and $x$ is any fixed object.