"Near equivalence" of semigroups and monoids

Question

Given any semigroup $S$ we can uniquely extend it to a monoid $M$ by introducing a new identity element (even if $S$ already has an identity we can still add a new identity). Conversely, given a monoid $M$ we can forget it's current identity to get a semigroup $S$, and these two operations are inverse to eachother. However this is not a functor, so we cannot really call semigroups and monoids equivalent. However it feels like anything done in a semigroup can be done in a monoid and vice versa using this operation. Is there a name for this "near equivalence"?