(OP)
Expressing leftist ideas to ancaps -
Something that I gotta mention first: open up their mind to the idea that they could be wrong. This is fucking important. Arrogance of any kind of either side drives meaningful discussion away fast. And this goes for you, too, but since you two are friends, this might not be an issue.
Also: AGREE UPON A DEFINITION OF CAPITALISM, SOCIALISM, AND ANY OTHER POSSIBLY AMBIGUOUS TERMS. Get the distinction of private and personal property out of the way along with the other terms asap. If you can't even get this much down, you'll never be able to get to this person. This may not be a problem in your specific case, but I've seen so, so many people mindlessly tossing thinking they're tossing shit at each other when in all reality they're actually just hitting their imaginary friend that's standing right next to them. Define your shit so each of you know what the fuck you're talking about. Moving on.
Question them about the absolute basis of what an ideology's goals should be. That gets right to the heart of it all, tbh. No bullshitting there. To leftists, it's something like "to distribute resources and create policy in a way that creates overall improved living standards and autonomy". Some ancaps would agree with this as the foundation of an ideology. If not, then they probably have a more individualist thing in mind, in that case, redirect them to Stirner. As memey as this guy may be, I've heard that he makes arguments for the left entirely based on the idea of individualism. I've never read him, but it's a guy that many people suggest when appealing to this kind of thing.
Whenever corporatism is mentioned: listen closely. Whenever this argument is raised, there's self-contradiction everywhere. Read part 2 of chapter 1 of Kropotkin's "Conquest of Bread". It talks about the idea of private ownership, but I'll interpret it slightly differently (for the sake of this context) and say that process of production and that it is never just from the manufacturing to the purchase. Every little thing goes much deeper than that. Every aspect of everybody's life today, everything that has ever been produced in the current capitalist market, has been effected by the state, whether that be for better or (more likely) worse. To say that you can point to anything and say "see now THAT'S capitalism! the state had no involvement in that! no corporatism there!" is absolutely absurd. Then you can maybe get into how the state and capitalism depend on one another, as they do. Something to remember: this is one of the beliefs that sits at the very core of their perception of the world today. It's what allows them to praise only what they see as positive aspects of the capitalist system today while shunning the parts they don't like by externalizing it then labeling it as something completely different. If you can get them to see why this is wrong, then it might, just might, be a downward slope from there. So call them out on their shit.
Ask them about monopolies. They be like "state does it". Google "sources of monopoly power". See that there are uh… one or two(?) valid arguments on the list on why monopolies could be held up by the state. What you need to focus on is what they don't bring up. It's the things that aren't caused by the state. That might get them to scratch their heads a bit. Might need to get to know the numbers on private subsidies so you can shoot down that shit when they might pull it up.
Remember, they're anti-state. Appeal to that. Ask them why they think the state is fucked up, the inherent problems that come with it, and then relate it back to capitalism when you feel you can draw a solid line of comparison.