I do it like @kakinami, push down while pulling but also on moving back of the tensioner. But the crosses still tend to point a little bit upwards and the mains inwards. I straighten both of them directly after stringing.
At tournaments, when I’m in a hurry, then only the crosses. But I still check quickly the mains to see if one has moved a bit more.
For me, not straightened strings are just as bad as crossovers.