Question on double pawns.

I am a beginner. I have a question on the double pawn structure. This was from a daily tournament game(elo 1200 -1400 only). I was black, and it was black's turn. During the game I did Bxe3, then white fxe3. We exchanged the bishops. I did so for the following two reasons: 1) This creates double pawns for my opponent, which is supposed to be good. 2) This opens up the f-file of my opponent, so the king side is no longer safe to castle. The opponent had moved c pawn, so it seemed to me there was no safe place for white to castle. However, there were two drawbacks too. 1) Color wise, I traded my good bishop with opponent's bad bishop. 2) My opponent got more pawns in the center. If he/she can handle it well, after some pawn exchange, the double pawn can be solved and transferred into pawn advantages in the center.

After the game, I analyzed the game using an engine. The engine told me not to exchange the bishop, but do d6 instead. I can kind of understand this, because there were 2 pros and 2 cons. It might not be worth it. However, if I were white, I will immediate do Bxc5 to exchange the bishop after black d6. For the following reasons: 1) to create double pawns for the opponent, 2) to exchange a bad-color bishop with a good-color bishop for a gain, and 3) move an opponent's pawn from center towards edge. There were only pros and no cons. However, the engine doesn't do the exchange, and does the Nf3 instead.

I am confused by the engine's move here. Why doesn't the engine do the bishop exchange here? I am also confused about the double pawn. Probably it is not as bad as I thought? Can anyone answer my question?

Thank you!

Thank you very much for your answer. I still have some questions. Does this mean that double pawn in the center may not be a bad thing, as long as it helps to control more center squares?

"White has the semi open f-file."

Is semi open file a good thing? Yes, it is easier for rooks, but you get weak defense too.

"Bc5 isnt good because it gives wite a backward d-pawn, and a target for black to attack."

Without Bc5, the backward d-pawn is still there.

After Bxc5, f4 becomes weaker. Also a pawn on c5 has much stronger control of d4 than a bishop there does. The white king will be plenty safe with a king side castle especially with the c5 bishop gone. Yes, semi open files are very good.

Thank you!

Does this mean that double pawn in the center may not be a bad thing, as long as it helps to control more center squares?

A weakness is only a weakness if it can be attacked.  So while doubled pawns can be a weakness, it usually shows in the endgame.  In the middlegame, you can use then to gain space, and control squares.

"White has the semi open f-file."

Is semi open file a good thing? Yes, it is easier for rooks, but you get weak defense too.

Rooks belong on open files.  This allows you to swing your rook over to the g-h files.

"Bc5 isnt good because it gives wite a backward d-pawn, and a target for black to attack."

Without Bc5, the backward d-pawn is still there.

Correct, but now your backwrd d--pawn cant be attacked because the d-file is not open.

@FishEyedFools

Thank you very much! Now I understand the topic much better!