  In the present article, we review the classical covariant formulation of
Yang-Mills theory and general relativity in the presence of spacetime
boundaries, focusing mainly on the derivation of the presymplectic forms and
their properties. We further revisit the introduction of the edge modes and the
conditions which justify them, in the context where only field-independent
gauge transformations are considered. We particularly show that the presence of
edge modes is not justified by gauge invariance of the presymplectic form, but
rather by the condition that the presymplectic form is degenerate on the
initial field space, which allows to relate this presymplectic form to the
symplectic form on the gauge reduced field space via pullback.
