Partner expecting things in return for doing housework

[Motherofone25]

My partner and I have been on rocks recently and the main topic is due to him not helping around the house and with our 4 month old.
we works full time and I’m at home on maternity. I clean and tidy the house daily but leave the heavy lifting tasks to do when my partner is at home to do together.
He is complaining that he has no chill time when he gets home and doesn’t like being asked to do things. He goes to the gym 4 times a week for 1.5hrs a time so he’s only home for 2-3hrs those days before going to bed.
I ask if he can do the last feed and nappy change before bed so he is still spending time with our child but he moans and complains about it.
he asks me what will he get in return? He is always seeking out sexual favours from me, since having our baby my sex drive has plummeted and we have spoke about this and I thought we were of an understanding that it’ll take time for me to get back to pre pregnancy body and mind. However, he believes he shouldn’t be doing any favours for me (his responsibilities he signed up for when moving into his own home and becoming a dad) unless I give him something in return, which is sex in his case.
not sure how to go about this situation as I respect he’s working full time and doesn’t get all of the down time he wants but his expectations are of someone that has no responsibilities of being a partner or dad.

When is your downtime? How often you go to the gym or anywhere on your own? Does he think maternity leave is a holiday? That you don't work all day?
The old bias of men's work being more important then woman's.

He told exactly where he thinks he has you. That you can't leave so he can do what he wants. This is horrible basis for relationship or family and it will only get worse unless you make it very clear to him it has to change and be prepared to follow with consequences if it doesn't.

What a horrible person he is.
Our first child came as a surprise and before that it was adamant I never wanted kids as I wanted my own time.
After she arrived everything changed. I did everything I could to have as much time with her as possible. Night changes and feeds, any nappy that needed changing. Playtime on the floor. Read stories every single night until she was too old (12 or so)
I never had to question what I got out of it. I got to be a dad and there is nothing better than that!
Our son was sat on the sofa yesterday playing and just burst into song about me. He's happy, our daughter is an amazing young women and confident and happy.
Thats what we did together. No other reward needed.
Also as a side note. Sex is for you both. Not as a reward for him ever. I hate the thought of that and could not imagine sleeping with my wife if she didnt want to.