We combine experimental investigation with numerical simulation to explore fundamental hydrodynamic effects of an inner droplet on deformation of a double emulsion droplet under shear. The transient deformation oscillation is found to be intensified by the inner droplet. Especially, we demonstrate that the double emulsion droplet can exhibit both larger and smaller steady deformation than the single-phase droplet, which arises from the competition between the coexisting enhancing and suppressing effects by the inner droplet on the deformation. We further provide a regime diagram to quantitatively recognize the respective dominant regime of these two effects, depending on the capillary number and radius ratio of the inner droplet to the outer one.