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In this sequel to works D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2)(arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), were-examine --- and reformulate when in need --- several basic notions in super$C^{\infty}$-algebraic geometry as guided by the mathematical formulation ofRamond-Neveu-Schwarz fermionic strings and of Green-Schwarz fermionic stringsfrom the viewpoint of Grothendieck on Algebraic Geometry. Two theorems that arethe super counterpart of Theorem~3.1.1 and Theorem~3.2.1 of D(11.3.1) areproved. They unify the notion of "smooth maps from an Azumaya/matrix supersmooth manifold with a fundamental module to a super smooth manifold"introduced in D(11.2), making it a complete super parallel to the setting forD-branes in the realm of algebraic geometry in D(1) (arXiv:0709.1515 [math.AG])and D(2) (arXiv:0809.2121 [math.AG]), and in the realm of differential or$C^{\infty}$-algebraic geometry in D(11.1) and D(11.3.1). A prototypicaldefinition of dynamical fermionic stacked D-brane world-volume on a space-timein the same spirit of RNS fermionic strings or GS fermionic strings is thuslaid down. Similar to D(11.3.1), which paved the path to the construction ofnon-Abelian Dirac-Born-Infeld action (D(13.1) (arXiv:1606.08529 [hep-th])) andthe standard action (D(13.3) (arXiv:1704.03237 [hep-th])) for fundamentalbosonic stacked D-branes, the current notes shall serve the same for theconstruction of supersymmetric action for fundamental fermionic stackedD-branes of various dimensions --- a theme of another subseries of theD-project. A notion of "noncommutative $C^{\infty}$-rings" and "morphism"between them is introduced at the end as a byproduct.

Further studies of the notion of differentiable maps from Azumaya/matrix  supermanifolds I. The smooth case: Ramond-Neveu-Schwarz and Green-Schwarz  meeting Grothendieck