For the first fitting, or for a toile, I usually ask for both sides the vent to be basted back on the CB line, so I can judge whether the backs are hanging correctly- if they are, the two edges will hang straight edge to edge. If they are crossing over in the back, or pulling apart I can easily see it and correct it at the fitting.
The vent from the outside looks like the photo on the left. The left side of the centre back is sewn to form a "step" that sits on top of the right back.
The middle photo shows the inside of the coat as the lining is being put in.
You need to clip the seam allowance on the left back to form the step vent but we don't clip the seam allowance on the right back as it makes that point much weaker. We press open the seam allowance from the neck down to a point a few inches above the vent, then gradually allow the seam allowance to fold and lay all in the same direction.
A note about the back lining ...for years, I always cut the back lining in one piece, but it is a more complicated technique to install it, so now it is cut in two pieces and it will have a short horizontal seam in the end. It makes much more sense to do it this way because it saves time, is stronger in the long run, and is easier to put in.
I confess that I have to be reminded everytime to cut it like this, because my automatic tendency is to do what I've always done. I just need a few more years practice for it to become my new "normal".