Its all about the lighting differential. The bigger the difference, the slower the process. The idea of "putting on the bottom" and slowly moving it up only makes sense if the frag is used to a lower light and moving to a final placement with higher light. This also comes in 2 forms - 1) frag that grew up in high light tank but spent 2 to 3 days in shipping box in total darkness, then moved back to tank with similar lighting. These only need a couple days to acclimate. 2) frags that grew up in low light but going to tank with much higher light. These acclimate take much longer. I typically go with a 5 par per day rule. Meaning, if the two tanks are 100 pars apart. I acclimate over about 3 weeks.
So as you can see, its not about just blindly following steps, more important it is to know the relative difference in par between the two tanks. Since it sounds like you got the frag right away and didn't spend time in a black box, you dont need to follow the 1) version of the acclimation and only follow 2) to the extent your light is stronger than your friends light.
Ie, for me, I know my LFS keeps acro frags under 400 to 450 Par. So, when I get them I glue them directly to parts of my tank that match the PAR. So far I have no loss in both coral or color following this method.
Only other variable is photoperiod but most of the time its similar. However of the two tanks are very different, you will probably want to compensate for longer hours with lower par. Ie, if both tanks are 300PAR but source one runs peak for 4 hrs and the destination one runs for 12, in that case you'd want to keep it lower level and slowly move it up to the 300 par height.