The answer to your question to your question is a bit to complicated to set out in an email. Have a look at Wooldridge (2010). Econometric Analysis of Croos section and Panel Data, MIT Press, Section 10.2.2 on page 287. Given stochastic y and x and unobserved effect c then the exogeneity condition is
E[y_{it}| x_{i1}, x_{i2},..., x_{iT}, c_i] = E[y_{it}| x_{it}, c_i] = x_{it}\beta + c_i
i.e. once one adds the unobserved effect the requirement is that the "exogeneity" should go if the fixed effects estimator is OK. At least this is what I understand to be the case.
Best Regards
John