Current trading is going to be more relevant than IPO comps - yes, you could show, but, firstly, IPO comps are going to be rarer than M&A, because IPOs are rarer.
How to value a company using discounted cash flow (DCF) - MoneyWeek Investment Tutorials
Secondly, current public multiples are more relevant than past multiples (which is essentially what a prior IPO shows). You could make an IPO discount point here, but that is something the ECM guys will go by experience on rather than actually quantifying.
DCF is generally included, but unless a company is trading at insane multiples, it is always going to be the highest valuation.
At the end of the day, you already know what value you think the company can go for, so are going to try to triangulate to that point, and are going to use whatever metrics are necessary (and make them up, if need be).
Components of an equity book would still vary, but let's say it's an IPO pitch - you'd probably have marketing pages, sector/market update, investment case for the company, valuation, key investors in the sector/country, proposed timeline, etc.