Questions to ask / things to diligence when being offered equity

Hi Folks,

Seeking some advice from this forum on comp.

For background:

I'm an ASO2 in an infra group at a BB. I like the sector and have received good reviews / developed a good reputation, but have for a while now felt that IB (and specifically my group) is not for me long-term, so have been looking to jump.

I got an offer this past week to join a young-ish infra investing shop focused on a specific sub-sector that I find interesting and think has a ton of growth. The role would be helping grow one of the firm's main platform investments and would involve a mix of M&A / investing and operating. The senior people across the firm (partners, as well as board / advisors) all come from really impressive investing or operating backgrounds and think it would be great to learn from them and throw my lot in with them.

The challenge is that the initial offer is a significant step-down in cash comp from what I'm making now / expect to come February and didn't include equity. They claim their hands are tied on cash comp, and ultimately I can make this number work in the short-term (though will not be thriving in NYC), but I told them I am interested in equity and am fairly sure their counter will include. Before they do I wanted to ask this forum what I should be considering / asking as I evaluate and potentially negotiate:

Here are some points to DD / questions I've come up with, please add or put a finer point to any of these:

1. Seniority - ultimately I want to be pari with and have incentives aligned with the decision makers

2. Current cap structure - they have an existing investment from a MM shop; I don't have much detail on this, but in my current role I see a lot of heavily structured investments with PIK and / or liquidation prefs so want to be clear on the range of outcomes here

3. Exits / liquidity event

4. Obvious stuff like amount, vesting schedule

Much appreciated.