Exercise 5.1.36

Let be a diagram and a limit cone over .

Theorem: If are maps such that for all , then .

Proof: Consider that is a cone over . Then, there is a unique map such that . Taking and both make this equality true (the former trivially and the latter by supposition), so we must have since is unique.

Consider the case where is the two-object discrete category, , and . Interpreting the theorem above in this setting amounts to choosing a pair of sets , and two elements of their product . The theorem then says that if two elements of have the same left entry and right entry, then the two elements are equal as tuples – in other words, equality of tuples is equivalent to component-wise equality.