Derivative of basis vector in terms of Christoffel symbols

The usual definition of the Christoffel symbols is $$nabla_mu {bf e}_nu = {bf e}_sigma {Gamma^sigma}_{numu} $$ where $nabla_mu $ is a covaraint derivative and the ${bf e}_mu$ are basis vectors of the tangent space $T(M)$. Your formula differs in sign for some reason. Where does it come from? Are the ${bf e}_a$ an orthonormal basis, and if so what are the ${bf e}^a$? Are they a coframe? If so, the sign is right, but they are not the basis vetors of the tansgent space, but rather of its dual $T^*(M)$. Also if this is the case then for a vector $X= X^a {bf e}_a$ we us the fact that evalauting a covector on a vector returns its components, i.e. ${bf e}^a(X)=X^a$, to see that $$ {bf e}_a {bf e}^a(X)= {bf e}_a X^mu=X. $$ Thus $sum_a {bf e}_a {bf e}^a$ is the identity map from $T(M)to T(M)$. No "$cdot$" is needed between the ${bf e}_a$ and the ${bf e}^a$.