**Exercise 2.1.15**

**Theorem**:
Left adjoints preserve initial objects

*Proof*:
Let and be functors with .
Let be an initial object.
Let be any object of .
Then, we have .
Since is initial the right term has exactly one element, which means the left one does as well.
Since was arbitrary, this means there is exactly one arrow for any , and so is initial in .

A similar argument shows that right adjoints preserve terminal objects.