According to the New York Times, the compromise is an attempt to "make the new rule more like that offered by the State of Hawaii, where employees of religiously affiliated institutions obtained contraceptives through a side benefit offered by insurance companies." However, they explain somewhat helpfully: "The result differs from Hawaii in that it shifts the cost to insurers, instead of employees. It also differs from Hawaii in that it requires companies — and not the religious institutions — to inform employees about how to arrange coverage."
I'm having trouble understanding what this change amounts to. It sounds like religiously affiliated employers
will no longer have to pay directly for employee's contraceptives: instead, they will only have to do so indirectly, through the insurance they will be forced to pay for. Under both arrangements, individuals (as opposed to institutions) are compelled to pay for other people's contraceptives via the insurance premiums they are forced to pay.
I'm also having a huge problem seeing how this is an improvement. It sounds like what the administration is thinking is that the original objection was that Catholic institutions are holy and can't get any birth control cooties on them. If the connection is sufficiently indirect, that's not too icky and so everyone can be fine with it.
The real issue is similar to the one involved in this paragraph from Thomas Jefferson's Virginia Statue of Religious Freedom (one of the three achievements he asked to be named on his epitaph):
That to compel a man to furnish contributions of money for the propagation of opinions which he disbelieves and abhors, is sinful and tyrannical; that even the forcing him to support this or that teacher of his own religious persuasion, is depriving him of the comfortable liberty of giving his contributions to the particular pastor whose morals he would make his pattern, and whose powers he feels most persuasive to righteousness; and is withdrawing from the ministry those temporary rewards, which proceeding from an approbation of their personal conduct, are an additional incitement to earnest and unremitting labours for the instruction of mankind.
Forcing someone to actively participate in an activity that they sincerely believe is wrong is morally problematic. So is forcing them to pay for advocacy of the idea that this activity is in fact not wrong (the case that Jefferson is addressing here). I would also put forcing someone to pay for carrying out this same activity belongs in the same category.