Compactness of the smooth Zariski stack coefficients

Let $(F_a)$ be a small filtered diagram of sheaves and put $F(Y)={\mathrm{colim}}_a{F}_a(Y)$ in spaces. Fix a test scheme $Y$. By [F2] its ordinary truncation is qcqs with the same open subsets. Its quasi-compact opens are closed under finite unions and intersections by [F1], and the associated derived opens remain in the smooth site by [F2]. Here a compact open means a quasi-compact open; coherent means quasi-compact with a basis of such opens closed under finite intersections. On these opens every $F_a$ sends the empty open to a point and satisfies $F_a(U\cup V)\simeq F_a(U){\times }_{F_a(U\cap V)}{F}_a(V)$. Filtered colimits preserve the finite limits in these conditions, so the same holds for $F$. The compact-open criterion extends this restriction uniquely to a sheaf on all opens of $Y$. Every Zariski cover of a qcqs object has a finite refinement by quasi-compact opens, since these form a basis and the object is quasi-compact; their finite intersections remain quasi-compact opens and objects of the smooth site. Consequently the just-verified sheaf condition on these opens is exactly descent for the covering sieves on the site. Thus $F$ is a site sheaf. This proves objectwise computation of the filtered colimit. The proof concerns ordinary descent, not hyperdescent.

By sheafification adjunction and Yoneda, $\mathrm{Map}(h_Y,F)=F(Y)$; step 1.1 therefore makes each $h_Y$ compact. Yoneda generation expresses every presheaf as a colimit of representables. Applying the colimit-preserving sheafification to that expression gives the asserted generation of sheaves; the family is small in the fixed universe. Unpointed cotensor satisfies $(F^{{\mathbb{P}}^1})(Y)=F(Y{\times }_S{\mathbb{P}}_S^1)$, which preserves filtered colimits by step 1.1 since the product is smooth qcqs by [F2]. For a pointed sheaf $F$, pointed maps from $({\mathbb{P}}^1,\mathrm{\infty })$ over $Y$ give $(F^{({\mathbb{P}}^1,\mathrm{\infty })})(Y)\simeq \mathrm{fib}(F(Y{\times }_S{\mathbb{P}}_S^1)\to F(Y))$, the fiber being over the specified basepoint section. Step 1.1 computes both evaluations objectwise in filtered diagrams, and [F1] commutes the fiber with the colimit. Hence the reduced cotensor also preserves filtered colimits. Cartesian multiplicativity is already supplied in [A1]. These are precisely the two additional coefficient hypotheses, with no extra geometric restriction.