As suggested by GH, if one replaces (b) with the strengthened: (b)' for each infinite sequence $\\{f_n\\}$ in $A$ and each infinite sequence $\\{x_n\\}$ in $D$, if the iterated limits $$\delta=\lim_{m\to\infty}\lim_{n\to\infty}f_m(x_n)\text{ and }\gamma=\lim_{n\to\infty}\lim_{m\to\infty}f_m(x_n)$$ exist in the one-point compactification $\overline{K}$ of $K$, then $\delta=\gamma$; Then one can show that (a) and (b)' imply (c), with some modifications of the original proof of Bourbaki.

Yes, I think it is a variant of the Eberlein-Smulian on the space $C_s(X)$. I will formulate the question more precisely. Thank you very much!