Father responsibilities?

What responsibilities should a father have? I’m genuinely asking. Last night, i nursed my son to sleep (he is 4 months). I started at 7pm and he likes to cuddle and nurse for a while sometimes, last night it was an hour. I put him in his father’s arms, asleep, so I could pick up my daughter (different father) from my moms. (It was around 8pm). I got back around 8:45 and my boyfriend was screaming at me saying he woke up and it took him 30 minutes to put him back to sleep and I’m horrible at managing my time, I’m not fair to our son and I should be here. I am really wondering if I’m wrong. And if I am, what responsibilities should he have if I do everything for our son? I’m just confused... thank you!