  We compute the indecomposable objects of \dot{U}^+_3 - the category that
categorifies the positive half of the quantum sl_3, and we decompose an
arbitrary object into indecomposable ones. On decategorified level we obtain
the Lusztig's canonical basis of the positive half U^+_q(sl_3) of the quantum
sl_3. We also categorify the higher quantum Serre relations in U_q^+(sl_3), by
defining a certain complex in the homotopy category of $\dot{U}^+_3$ that is
homotopic to zero. We work with the category $\dot{U}^+_3$ that is defined over
the ring of integers. This paper is based on the (extended) diagrammatic
calculus introduced to categorify quantum groups.
