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In this follow-up of our earlier two works D(11.1) (arXiv:1406.0929[math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]) in the D-project, we studyfurther the notion of a `differentiable map from an Azumaya/matrix manifold toa real manifold'. A conjecture is made that the notion of differentiable mapsfrom Azumaya/matrix manifolds as defined in D(11.1) is equivalent to onedefined through the contravariant ring-homomorphisms alone. A proof of thisconjecture for the smooth (i.e. $C^{\infty}$) case is given in this note. Thus,at least in the smooth case, our setting for D-branes in the realm ofdifferential geometry is completely parallel to that in the realm of algebraicgeometry, cf.\ arXiv:0709.1515 [math.AG] and arXiv:0809.2121 [math.AG]. Arelated conjecture on such maps to ${\Bbb R}^n$, as a $C^k$-manifold, and itsproof in the $C^{\infty}$ case is also given. As a by-product, a conjecture ona division lemma in the finitely differentiable case that generalizes thedivision lemma in the smooth case from Malgrange is given in the end, as wellas other comments on the conjectures in the general $C^k$ case. We remark thatthere are similar conjectures in general and theorems in the smooth case forthe fermionic/super generalization of the notion.

Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case