Forum adverts like this one are shown to any user who is not logged in. Join us by filling out a tiny 3 field form and you will get your own, free, dakka user account which gives a good range of benefits to you: No adverts like this in the forums anymore. Times and dates in your local timezone. Full tracking of what you have read so you can skip to your first unread post, easily see what has changed since you last logged in, and easily see what is new at a glance. Email notifications for threads you want to watch closely. Being a part of the oldest wargaming community on the net. If you are already a member then feel free to login now.

If an Arch Lector mounted on the War Altar casts his battle prayer and a regimental unit is within 6", while it's detachment is not (yet still w/in 3" of the regimental unit), does the detachment benefit from the prayer?

If it's one of those abilities that cascades down through detachments, sure.

As above.

If the parent unit gets one of the abilities on the list, no matter the source, the detachments will get the rule too.

Awesome! I assume that also applies to the Hatred conferred by the War Altar, especially as that is a form of Psychology adopted by the regiment.

Only if Hatred is on the list of stuff that applies down to the Detachments.

it is, as long as detachments are within 3" of the parent unit, they get any/all bonuses that the parent unit receives, regardless of source. (as long as it is one of the transferable bonuses)