Map object versus product
The problem of finding map objects in a given category is complicated by the fact that often the map object we are looking for does not exist. This difficulty occurs many times in mathematics: we have a problem and we do not even know whether it has a solution. In such cases it is often helpful to pretend that the problem does have a solution, and proceed to calculate anyway! We need an account of
how to use wishful thinking
which we will then apply to the problem of determining map objects in the category of sets and in the category of graphs.
We imagine that we have already found the solution to a given problem, and try to deduce consequences from its existence. We ask ourselves: What does this solution ...