L et X be a locally arcwise coıuıected topological space. In paper [1 ], we constructed “ the sheaf of the fundam ental groups” on X and gave some characterizations. In paper [2], we first gave some characterizations which are th e converses of the characterizations in paper [1 ] and obtained some results related w ith the group of sections. In this paper, we first construct the sheaf H* = H ı © H2 by defining “ the Generalized W hitney Sum ” and show th a t the sheaf H* is isomorphic to the sheaf HJXH2 which is direct product of the sheaves and H2 on X j and Xs, respectively. Finally ,we prove th a t, if (X j, H ı) and (X2, H2) are any two pairs, th en H* is iso­ m orphic to the pairs (H, X ı xX2).