Although they appear quite different, Gibbs sampling is a special case of the Metropolis-Hasting algorithm
Specifically, Gibbs sampling involves a proposal from the full conditional distribution, which always has a Metropolis-Hastings ratio of 1 – i.e., the proposal is always accepted
Thus, Gibbs sampling produces a Markov chain whose stationary distribution is the posterior distribution, for all the same reasons that the Metropolis-Hastings algorithm works
Reference:
https://web.as.uky.edu/statistics/users/pbreheny/701/S13/notes/2-28.pdf
