The same thing happens with our circle of fifths if we move counter-clockwise, to the left, from C as a starting point.
We start at the key of C major (or A minor, which is C major's relative minor). The C major scale, and therefore the key of C major, has all natural notes (c, d, e, f, g, a, b) and therefore zero sharps or flats.
If we move down a fifth from C we arrive at the note F. F is down a fifth from C.
Applying the major scale formula starting at the note F results in the notes of the F major scale, and therefore the key of F major (f, g, a, b-flat, c, d, e) and therefore it has one flat.
We move down again a fifth and arrive at the note B-flat. B-flat is down a fifth from F. Applying the major scale formula starting at the note B-flat results in the notes of the B-flat major scale, and therefore the key of B-flat major (b-flat, c, d, e-flat, f, g, a) and therefore it has two flats.
If we keep working in either direction we continue to add one sharp or flat to the key signature each time.
Eventually, we wind up at the bottom of the Circle of Fifths where the sharps and flats meet at the key signatures of F-sharp and G-flat. These two are the same thing with different names (enharmonically named).
If we continue on and move back around to the top in either direction we find that are still moving in fifths, but the nature of the circle of fifths, the music notes and the efficiency of the entire system is such that they flow right into one another!
Moving from the F-sharp up a fifth results in C-sharp otherwise known as D-flat. Moving from the G-flat down a fifth results in C-flat otherwise known as B.
The entire thing is integrated and self-contained.
Here is the complete order of key signatures on the Circle of Fifths.
Notice that we wind right back a C where we started. Notice also that the flats are added in fourths, which are an inversion of the fifth intervals. Moving in the direction of the flats is sometimes referred to as a Circle of Fourths.