We're in Rome. The Cardinal is talking to Castruchio's wife Julia, saying something along the lines of, "so, what did you tell your husband this time to get yourself out here?" Wait a second. Ew.
Yes, that's right. Julia and the Cardinal. Together. Doing things. Things that are not playing chess or discussing Plato or praying. Yeah.
They get into a discussion about infidelity. Topical, no? Julia's already a cheater because she's having an affair with the Cardinal, but the Cardinal says that she'll be unfaithful to him, too. Gee, what a keeper.
A servant comes to tell Julia that Delio and Castruchio have both come to Rome, and that Delio's waiting outside to see her.
The Cardinal leaves, and Delio comes in.
Turns out Delio used to have a thing for Julia, and the first thing he does is offer Julia money.
Wait, what? This is weird. Julia doesn't know what's up, until Delio tells her he'd really appreciate it if she'd become his mistress. You know, when she's not with Castruchio. Julia's either got Victoria's Secret levels of hotness or she's just exceedingly politically convenient.
It's not quite clear on why Delio is doing this, but our best guess is that he's trying to get in with Julia so he can keep tabs on the Cardinal and help protect Antonio and the Duchess.
Julia's admirably quick on the draw: "Sure, you know what, Delio, I'll get right back to you; lemme go ask my husband what he thinks."
She leaves Delio alone, who's getting really worried. He hears that the Cardinal is enraged about some letter he's received from Malfi, and fears that somehow Antonio's marriage to the Duchess has been revealed.