= Solution
For coordinate $i$, keep the other coordinates fixed and propose $y_i$ from the full conditional $\pi(\cdot\mid x_{-i})$. This is a <Metropolis–Hastings algorithm> proposal on that coordinate fibre, with deterministic equality $y_{-i}=x_{-i}$. Its acceptance ratio is
$$
\frac{\pi(y_i,x_{-i})\pi(x_i\mid x_{-i})}{\pi(x_i,x_{-i})\pi(y_i\mid x_{-i})}=1.
$$
Thus the coordinate update has \b[acceptance probability one] and is exactly a single-coordinate <Gibbs sampler> update.
A systematic sweep through coordinates $1,\ldots,p$ is the composition of these kernels. Its joint transition density is
$$
\boxed{K(x,y)=\prod_{i=1}^p\pi(y_i\mid y_1,\ldots,y_{i-1},x_{i+1},\ldots,x_p).}
$$
The already updated coordinates are new values, and the not-yet-updated coordinates are old values. Each coordinate kernel preserves $\pi$, so their composition also preserves it. <Systematic Gibbs sampling need not be reversible>: each individual coordinate kernel is reversible, but their ordered composition need not be reversible; invariance is the property needed here. A random-scan mixture of the coordinate kernels is reversible.
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