Is my understanding correct?

Sir, I’m not the most technically compatible when it comes to subjects like AFM, hence I like to conceptually understand a topic and then memorize my understanding. Therefore I’m asking this question so that you may, if you can, validate if my understanding is correct.

For cost of equity: (the return that equity holders want)

If they give us a competitors equity beta, we calculate their asset beta using the formula. We do this because we want to extract the business risk and not their financial risk (because our financial risk is different). Key thing to note here is that, when we use the formula we are using their market values of debt and equity.

We then use their asset beta, and assume it is same for us. This is because we assume that since we operate in the same industry (us and our competitor), therefore it is appropriate to assume it is the same. We then use this asset beta and use the same formula to calculate the equity beta. But this time around, we use our market values of debt and equity, we do this so that we find a representation of our business and financial risk.

We then use this equity beta to calculate cost of equity using the CAPM formula.

Also,
Asset beta : represents business risk, hence the reason for isolating it in the first place
Equity beta: represents financial risk and business risk, hence the reason for using both our market values in the second step.

Is all of this conceptually correct? Thank you!!