I have this westcott apollo, the end I believe is not tapered, it is shaped just like a normal umbrella. But I am not sure, I can go back home and check it for you. I am not sure how big is your equipment, but I am able to squeeze a bowens monolite 150 inside, its the smaller monolight. There are 2 models for this, one is built for monolights and one is built for hotshoe flashes. I have the hotshoe flash version. One problem with squeezing a bigger monolight into the hotshoe flash version is that there is quite a large hotspot.. light is still diffused but because the monolite's bulb is so much more forward compared to the hotshoe flash, light does not disperse as efficiently. Hope this helps. If you shoot exclusively with monolights I suggest just buying the monolight version.
Thanks. I hold a profoto AcuteB lights with the umbrella holder built into the light. There is a ball bearing in the light that hold the umbrella. The shaft end must be tapered in order to go through. Or else I need to file the end.