Given two objects $A,B$ in Set with an arrow $A\xrightarrow{f} B$ between them, find the limit and colimit of this diagram.
By definition, the limit is the unique universal left solution, that is, it is an object $X$ in Set with arrows $X\xrightarrow{s_A} A$ and $X\xrightarrow{s_B}B$ making the appropriate diagram commute (I'd draw it but I'm not sure how). We must also have that for any other left solution $S$, there is a unique arrow $X\to S$, again making the correct diagrams commute.
With this in mind, I think that the limit of the diagram should be the null set, since there is certainly a unique map from the null set into any other possible solution, and commutivity would be trivially satisfied. However, I'm not really sure what the colimit should be. I think it might be the power set of $A\cup B$, but I'm not really sure how to show this. Any suggestions?