Orbit sets, fppf quotient sheaves and representing schemes are three different objects

Assume the Axiom of Choice for the represented-sheaf and geometric suppliers used on this page. Three objects are commonly conflated, and must be kept apart. (1) The orbit set $U(k)\mathrm{/}R(k)$: the value at $k$ of the naive quotient presheaf of a pre-relation; it is computed from $k$-points alone. (2) The fppf quotient sheaf $U\mathrm{/}R$ (Quotient sheaves and representable quotients for pre-relations and group actions): the sheafification of that presheaf, whose sections over a $k$-scheme $T$ are fppf-local orbit data. Its $k$-points can be strictly larger than the orbit set, and it carries infinitesimal information invisible to it; a concrete instance is recorded on the examples companion page of this pair. (3) A representing scheme: a $k$-scheme $M$ with a natural isomorphism $h_M\cong U\mathrm{/}R$ (Criterion for a scheme to represent an fppf quotient sheaf); by Yoneda it is unique up to unique isomorphism when it exists. Representability is an additional property, not a consequence of the definitions: this page establishes it for homogeneous spaces of smooth affine groups (Homogeneous spaces of smooth affine groups are separated schemes) and for affine finite locally free equivalence relations (Affine finite locally free equivalence relations have finite locally free scheme quotients). In the homogeneous-space case the quotient is the orbit of the base point and the quotient morphism is faithfully flat (A faithfully flat orbit map represents the coset quotient sheaf); the general arbitrary-group quotient representability statement is outside the scope of this page and is not claimed here.