How does Doomcaller interact with a polymorphed/hexed/entombed C'thun?

How does Doomcaller interact with a polymorphed/hexed/entombed C'thun?

Since technically a sheep or a frog dies, is it possible to place the lost C'thun back into the deck, or does Doomcaller do nothing?

Also, if it is in your opponent's deck due to a card like entomb, would it bring it back into yours? Or is it still technically "alive" and therefore unreachable by Doomcaller?

If a C'thun has died under your control this game, it will shuffle a C'thun into your deck.

Polymorph and Hex make it so that it is not a C'thun that died. Entomb does not have a C'thun die at all.

Incidentally, this also means the second copy of Doomcaller can shuffle back in a second C'thun, and Brann can cause a single Doomcaller to shuffle in two C'thuns. It also means that Steal Effects such as Mind Control will let you shuffle in additional C'thuns (once your opponent's dies) even if you haven't drawn your first yet, while simultaneously preventing your opponent from getting Doomcaller value.