Equivalences preserve small projective generators

(Set-up.) Equip $F$ with the adjoint-equivalence data $F⊣G$, $\eta$, $\epsilon$ of [F2]; then $F$ and $G$ preserve and reflect all existing colimits by [F3] and preserve epimorphisms by [F4], and the transposition ${\mathrm{\Phi }}_{c,d}$ of [F7] is a natural bijection for all $c\in \mathcal{C}$, $d\in \mathcal{D}$. The same properties hold for the equivalence $G$, whose quasi-inverse is $F$, with unit ${\epsilon }^{-1}$ and counit ${\eta }^{-1}$, and with transposition $\mathcal{C}(Gd,c)\cong \mathcal{D}(d,Fc)$.

(Forward: projectivity transfers.) Assume $P$ is projective, let $q:E\twoheadrightarrow M$ be an epimorphism in $\mathcal{D}$ and $f:F(P)\to M$. Then $G(q)$ is an epimorphism, and $G(f)\circ {\eta }_P:P\to G(M)$ is a map, so by projectivity of $P$ and [F5] there is $s:P\to G(E)$ with $G(q)\circ s=G(f)\circ {\eta }_P$. Put $t:={\epsilon }_E\circ F(s):F(P)\to E$. Using naturality of $\epsilon$ at $f$, naturality of $\eta$ at $s$, and the triangle identity ${\epsilon }_{F(P)}\circ F({\eta }_P)=1_{F(P)}$, one computes $q\circ t={\epsilon }_M\circ F(G(q)\circ s)={\epsilon }_M\circ F(G(f)\circ {\eta }_P)=f\circ {\epsilon }_{F(P)}\circ F({\eta }_P)=f$. Hence every epimorphism onto $F(P)$ splits, so $F(P)$ is projective by [F5].

(Forward: coproduct preservation transfers.) Assume $\mathcal{C}(P,-)$ preserves set-indexed coproducts. For every set-indexed family $(Y_i)$ in $\mathcal{D}$, the natural bijection [F7] and preservation of coproducts by $G$ give natural bijections $\mathcal{D}(F(P),{\coprod }_i{Y}_i)\cong \mathcal{C}(P,G({\coprod }_i{Y}_i))\cong \mathcal{C}(P,{\coprod }_{i}G(Y_i))\cong {⨁}_i\mathcal{C}(P,G(Y_i))\cong {⨁}_i\mathcal{D}(F(P),Y_i)$; all stages are natural in the family, so the canonical comparison ${⨁}_i\mathcal{D}(F(P),Y_i)\to \mathcal{D}(F(P),{\coprod }_i{Y}_i)$ is an isomorphism of abelian groups, since the adjoint transpositions are additive by [F4] and their formulas in [F7], which is exactly preservation of this coproduct.

(Forward: the generator condition transfers.) Assume $\mathcal{C}(P,-)$ is faithful, and let $u\mathrm{\ne }v:X\to Y$ in $\mathcal{D}$. Since $G$ is faithful (as part of the equivalence), $G(u)\mathrm{\ne }G(v)$; by faithfulness of $\mathcal{C}(P,-)$ there is $h:P\to G(X)$ with $G(u)\circ h\mathrm{\ne }G(v)\circ h$. Let $h^{\mathrm{♯}}:={\epsilon }_X\circ F(h):F(P)\to X$ be the transpose of $h$ under [F7]. By the formula ${\mathrm{\Phi }}_{P,Y}(w)=G(w)\circ {\eta }_P$ and naturality of $\eta$ at $h$, ${\mathrm{\Phi }}_{P,Y}(u\circ h^{\mathrm{♯}})=G(u)\circ G({\epsilon }_X)\circ GF(h)\circ {\eta }_P=G(u)\circ G({\epsilon }_X)\circ {\eta }_{G(X)}\circ h=G(u)\circ h$, and likewise for $v$, using the triangle identity $G({\epsilon }_X)\circ {\eta }_{G(X)}=1_{G(X)}$. Since ${\mathrm{\Phi }}_{P,Y}$ is injective, $u\circ h^{\mathrm{♯}}\mathrm{\ne }v\circ h^{\mathrm{♯}}$, so $\mathcal{D}(F(P),-)$ is faithful and $F(P)$ is a generator by [F6].

(Forward: the target is cocomplete.) Let $X:J\to \mathcal{D}$ be any small diagram and let $(L,\lambda )$ be a colimiting cocone of the composite diagram $G\circ X$ in $\mathcal{C}$, which exists because $\mathcal{C}$ is cocomplete; no choice is needed because the argument verifies any such cocone. Then $F(L)$ with the cocone $F\lambda :FGX\Rightarrow \mathrm{\Delta }F(L)$ is a colimit of $FGX$ by [F3], and the counit $\epsilon$ is a natural isomorphism $FG\Rightarrow 1_{\mathcal{D}}$, so the legs $F({\lambda }_j)\circ {\epsilon }_{X(j)}^{-1}$ give a colimiting cocone of $X$ with apex $F(L)$. Hence every small diagram in $\mathcal{D}$ has a colimit, and $\mathcal{D}$ is cocomplete.

(Forward conclusion.) Under the assumption that $P$ is a small projective generator of $\mathcal{C}$, steps 2.1, 2.2 and 2.3 show that $F(P)$ is projective, that $\mathcal{D}(F(P),-)$ preserves every set-indexed coproduct, and that it is faithful, while step 2.4 shows $\mathcal{D}$ is cocomplete; by [F1] and [F6] the object $F(P)$ is a small projective generator of $\mathcal{D}$.

(Converse.) Assume $\mathcal{D}$ is cocomplete and $Q:=F(P)$ is a small projective generator of $\mathcal{D}$. The argument of steps 2.1-2.4 applies to the equivalence $G:\mathcal{D}\to \mathcal{C}$, whose source $\mathcal{D}$ is now cocomplete, and shows that $G(Q)=GF(P)$ is a small projective generator of $\mathcal{C}$. The unit ${\eta }_P:P\to GF(P)$ is an isomorphism and induces a natural isomorphism $\mathcal{C}(GF(P),-)\cong \mathcal{C}(P,-)$, $\psi \mapsto \psi \circ {\eta }_P$; consequently faithfulness, preservation of coproducts, and the splitting characterization of projectivity transfer along it (for projectivity, transport a map $P\to M$ and its lift through ${\eta }_P$ and ${\eta }_P^{-1}$). Hence $P$ is itself a small projective generator of $\mathcal{C}$.

Steps 3.1 and 4.1 prove the two implications, and step 2.4 supplies the cocompleteness of $\mathcal{D}$ in the case where the properties hold; no commutativity of rings is involved and the only selections are the pointwise existentials supplied by the given lifting property and by cocompleteness, so no choice principle is used.