Wrong, GLUT4 is specifically for glucose. GLUT4 opens a gate on the plasma membrane so that glucose can be shuttled into the cell to form glycogen. Insulin suppressed beta-oxidation (oxidation of fatty acids) by directly inhibiting lipase activity. This stops the metabolism of stored fatty acids, it does NOT store fat.
GLUT4 is certainly found in both muscle tissue and adipose cells. A series of cellular phosphorylations (ranging from the phosphorylation of tyrosine residues and tyrosine kinase activity within cytoplasmic proteins, to insulin receptor substrate activation of phosphatidylinositol-3-kinase, to activation of GTP-binding protein ras, and ultimately to the mitogen-activated protein kinase system (MAPK)) induce several effects, including the translocation of glucose transporter proteins, especially GLUT4, to the cell membrane, leading to glucose uptake, higher glycogen formation, increased activity of the glucose synthase enzyme, activation of transcription factors leading to improved DNA synthesis, and so on. In fact, GLUT4 is one of the only GLUT proteins specifically present in adipose cells.
For those interested, GLUT1, for instance, is found in all tissues, especially in red cells, and in the brain. It is mainly involved in glucose uptake and transport across the blood-brain barrier. GLUT2 is mainly found in the liver, in the gut, in the kidney, and in the B cells of the pancreas, and is involved in the transport of glucose and fructose, as well as in glucose homeostasis, especially in the regulation of the release of insulin. GLUT3 is primarily found in the brain, in the placenta, in the kidney, and in some other tissues. It is involved in glucose uptake in neurons and other tisues. GLUT5 is found in the gut and kidney, and is involved in the absorption of fructose. While GLUT5 is not the only protein that transports fructose, it is the only GLUT protein that transports fructose.
Altogether, 14 GLUT proteins have been identified. Only 1-5 have been well characterized.