FCF & FCFE for valuation of an organisation/acquisition – confused

Good morning,
i wondered whether you would be able to help please,as i am quite confused regarding the use of FCF and FCFE for the purpose of valuing an organisation.

Here is what i believe i have understood so far:

we use FCF, which we discount at WACC to get to the value of a whole company (i.e debt and equity)

We use FCFE, which we discount at cost of equity, to get the value of the equity. (to value a company that would be fully financed by equity)

Now, going through the BPP textbook, what i can also gather is that when trying to value a target company (in the context of an acquisition), we can use both method i.e the FCF and the FCFE, to get to the equity value of the target company. (as long as we deduct the debt from the NPV when using the FCF method)

So what this tells me, is that to value a target company, i am only interested in valuing its equity, is that right? Because if this is right, i do not get why we only want the value of the target’s equity rather than the whole value (i.e debt+equity)?

Please can someone help?

Thanks 🙂