Factor Models 2: An Equilibrium Proof of the Capital Asset Pricing Model

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The derivative dr_a / dsigma_a at a=0, reflects addition of a very small amount of asset i to the portfolio (and deducing a very small amount of M from the portfolio as well), instead of the addition of the total market M (the exposure to new small risk disgma_a, even at a=0, is done by increasing a alittle). Hence, the new portfolio is not on the CML. So why would this derivative be equal to the sharpe ratio of M? I believe one should add an equilibrium statement as follows: Assuming a person in the CAPM world holds the tangency portfolio M, and considers whether to add to his portfolio a very small amount of asset i, or just another copies of M. At this position, by equilibrium consideration, the investor would buy asset i, only if the risk-return of this small addition to his portfolio M, would lead to the same sharpe ratio he could have been obtained by simply buying more stocks of M. Otherwise, the investor won't have the incentive to buy a small amount of asset i, and prices of asset i would fall. Hence, the equation is correct. Am I right?