The relationship for the James two sample test (see the function james.hotel) is true for the case of the MANOVA. The estimate of the common mean, \(\pmb{mu}_c\) (see the function james for the expression of \(\pmb{\mu}_c\)), is in general, for \(g\) groups, each of sample size \(n_i\), written as
$$
\hat{\pmb{\mu}}_c = \left(\sum_{i=1}^gn_i{\bf S}_i^{-1}\right)^{-1}\sum_{i=1}^gn_i{\bf S}_i^{-1}\bar{{\bf X}}_i.
$$
The function is just a proof of the mathematics you will find in Emerson (2009, pg. 76--81) and is again intended for educational purposes.