Under the basic assumptions on generators，the one-to-one correspondence between convexity and conditional convexity of

g-

expectations and generators of backward stochastic differential equations is obtained. Thus it is proved that the definitions of dynamic convex risk measures in Detlefsen-Scandolo (2005) and Jiang (2008) are completely same under the framework of

g-

expectations. Moreover

the one-to-one relation between time consistent dynamic convex risk measures via

g-

expectations and convexity of

g-

expectations is also established.