Am I doing the right thing?

Josh
I'm engaged to a beautiful woman and she has a child already been part of that child's life since basically birth and still is a baby he is only 20months he calls me daddy and I take care of him like he is my own son all my free time goes to him his dad is in the picture but he has very little time with him and he gets upset when he hear him call me daddy like for example when it's time for his visitation we drop him off or he picks him up he cries and screams for me yelling daddy daddy then his dad gets upset and tells him he is not daddy I am but he barely wants to spend any time with him he drops him back of early instead of asking for more time so I check him on that?should I tell him to step up or step down everyone tells me any guy can make a baby but it takes a man to raise one but the question that I wanted to ask is do I keep letting him run his mouth about me to him and putting me down if I'm doing his job and took his responsibilities