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What is the theory for this opening for black?
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
1. e4 e5 2. Bc4 c6
I've started playing the Bishop's Opening because it is something different, but I had not seen this counter play before.
It seems weak, because--although it has the advantages of an opening like Caro-Kann by grabbing control over the center--black immediately loses defense of his king pawn (Nc6 is unplayable). Despite this, it seems a solid opening for black.
Usually 2...Nf6 is played and only then 3...c6; this makes a difference. 2...c6 is rarely played and in response 3.Qe2 is strong.
With 2...Nf6 White is encouraged to defend his e-pawn. If 3.Qe2 Black can switch from 3...c6 to 3...Nc6/3...d6 lines and the Queen on e2 doesn't help ideas of playing for d2-d4. Now with 1.e4 e5 2.Bc4 c6 3.Qe2 Nf6, White can play c3 and d4 without loss of time.
Black cannot play Nc6 at the moment, that is true. However, it doesn't look like White can exploit this immediately, either.
Black will answer 3 Nf3 by ...d5, gaining tempo on the Bc4 that he can then use to either defend e5 (d7 is open for the knight, at the very least) or counter-pressure e4 (which is equally weak as e5). The whole motive is similar to the classical center attack in the Italian (c3 + d4).
Trading the pawns (4 exd5 cxd5) opens c6 for the knight again and doesn't look very attractive as it yields the center to Black. 5 Bb5+ is toothless, Black will reply ...Bd7 and then take back on d7 with his knight (or simply play Nc6 if the white bishop retreats now). Immediately retreating (by 4 Bb3) doesn't look very good either, Black will gain the initiative.
Of course, White has other options for his third move. 3 d4 is a similar attempt to attack e5. Black can reply 3...d5 again, which leads to a very complicated position. But again, e4 is just as weak as e5 and swapping away e4 opens c6 for the knight. No clear advantage for any side.
Against 3 Nc3 (prevents 3...d5, but doesn't attack e5), Black has time to play 3...Bc5 , followed by 4...d6 (e5 is safe) and he will achieve something similar to a Giuoco Pianissimo Reversed - which is not too bad, really.
So, yeah: Totally playable in my opinion.
Disclaimer: I'm not an actual expert in this opening.
2...c6 is one of Black's best replies to 2.Bc4, as it aims to play ...d5 and take advantage of White's bishop being developed so early. If White let's this happen, Black gains a big centre with tempo, and White loses any chances for an opening advantage.
As you noted though, the weakness of ...c6 is that Black cannot play 2...Nc6. Thus, the best way to play is to attack Black's e5-pawn (since Black cannot defend it with ...Nc6).
The theory isn't too developed, but here is how the main line runs (according to the MEGA database):
1.e4 e5 2.Bc4 c6 3.Nc3 Nf6 4.Nf3 d5 5.exd5 cxd5 6.Bb5+ Bd7 7.Nxe5 Bxb5 8.Nxb5 Qb6 9.Qe2 Be7 10.d4!
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
1.e4 e5 2.Bc4 c6 3.Nc3 Nf6 4.Nf3 d5 5.exd5 cxd5 6.Bb5+ Bd7 7.Nxe5 Bxb5 8.Nxb5 Qb6 9.Qe2 Be7 10.d4!
A few notes on the line:
By playing 4.Nf3, White forces Black's hand. Either Black must passively defend the e5-pawn with 4...d6, and then White may continue with 5.d4! Or, Black plays 4...d5, and White succeeds in provoking Black to take action.
White allows Black to get a big centre, but because Black is ill-prepared to defend his pawns, White wins the e5-pawn.
The ending position is slightly better for White. He's up a pawn, but it isn't much... White's bishop is bad and the knight on b5 is misplaced.
