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Progenitor Mimic copies exactly what was printed on the original creature and nothing more (unless that creature is copying something else or is a token). It doesn't copy whether that creature is tapped or untapped, whether it has any counters on it or Auras attached to it, or any non-copy effects that have changed its power, toughness, types, color, or so on.

The original Progenitor Mimic will create tokens, but those token copies will not.

If you choose to have Progenitor Mimic enter the battlefield as a copy of a creature, the triggered ability it gains will become part of its copiable values. For example, suppose Progenitor Mimic enters the battlefield as a copy of Runeclaw Bear, a 2/2 green Bear creature with mana cost {1}{G}. The resulting object is a 2/2 green Bear creature named Runeclaw Bear with mana cost {1}{G} and with “At the beginning of your upkeep, if this creature isn't a token, put a token onto the battlefield that's a copy of this creature.” If another Progenitor Mimic enters the battlefield as a copy of that creature, it will be a Runeclaw Bear with two instances of the triggered ability.

If the chosen creature is a token, Progenitor Mimic copies the original characteristics of that token as stated by the effect that put it onto the battlefield. Copying a token doesn't make Progenitor Mimic become a token.

Any enters-the-battlefield abilities of the copied creature will trigger when Progenitor Mimic enters the battlefield. Any “as [this creature] enters the battlefield” or “[this creature] enters the battlefield with” abilities of the chosen creature will also work.

If the chosen creature has {X} in its mana cost (such as Protean Hydra), X is considered to be zero.

If Progenitor Mimic somehow enters the battlefield at the same time as another creature, it can't become a copy of that creature. You may choose only a creature that's already on the battlefield.

You can choose not to copy anything. In that case, Progenitor Mimic enters the battlefield as a 0/0 Shapeshifter creature and is probably put into the graveyard immediately.