In this work, we consider area non-increasing maps between manifolds with positive curvature. By employing the strong maximum principle along the graphical mean curvature flow, we show that such maps are either homotopy trivial, Riemannian submersions, local isometries, or isometric immersions. This confirms the speculation of Tsai-Tsui-Wang. Additionally, we utilize Brendle's sphere theorem and the mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive 1-isotropic curvature. 