Reflective functors?

Question

Let $C_0\subseteq C$ and $D_0\subseteq D$ be reflective subcategories with reflection functors $r_A$ and $r_B$. For any functor $F:C\to D$, we may consider the natural transformation $r_BF\eta_A:r_BF\to r_BFr_A$.

Is there a name for those functors $F$ for which this natural transformation is an isomorphism?

If $F$ has a right adjoint $F^\ast$, then it is equivalent to ask that $F^\ast(D_0)\subseteq C_0$. This implies that the adjunction $(F,F^\ast)$ descends to an adjunction of functors between $C_0$ and $D_0$.

For example, sheaves inside presheaves $\mathrm{Shv}(X)\subseteq\mathrm P(X)$ is a reflective subcategory. If $f:X\to Y$ is a map of topological spaces, then we have an adjoint pair $(f^\ast,f_\ast)$ of pullback and pushforward functors on presheaf categories $f_\ast:\mathrm{P}(X)\to\mathrm P(Y):f^\ast$. The right adjoint pushforward $f_\ast$ preserves sheaves, so the left adjoint $f^\ast$ satisfies the condition above, namely that pulling back and sheafifying sends sheafifications to isomorphisms. This is how one proves that the adjunction $(f^\ast,f_\ast)$ descends to sheaf categories.

Answer 1

I don't have an answer, but an observation which might be relevant and is too long for a comment:

Consider categories $C,D,E$ with reflective subcategories $C_0,D_0,E_0$, and denote the reflections by $c\mapsto \overline c$. Given a functor $F:C\to D$, we obtain a functor $\tilde F: C_0\to D_0$ by $\tilde Fc=\overline{Fc}$.

We may ask whether this operation is functorial. In general it is only oplax functorial, with a generally non-invertible natural transformation $\gamma:\widetilde{GF}\to\widetilde G∘\widetilde F$ with components $$\gamma_c : \overline{GFc}\to\overline{G\overline{Fc}}.$$ Now $\gamma$ is invertible whenever your condition holds for $G$, i.e.

• ($\dagger$) $\overline{Gd}\cong\overline{G\overline d}$ for all $d\in D$.

This phenomenon of

• ($\star$) an oplax functorial construction which is strongly functorial whenever the second composee is in a special class of arrows

is characteristic of morphisms of equipments: an equipment is just a $2$-category with a designated subcategory of "special" $1$-cells (replete and containing equivalences), and equipments form a $2$-category enriched category (in the sense of Verity [1]), whose $1$-cells -- which I call "special functors" in [2] -- are the (weakly normal) oplax functors which preserve special $1$-cells and satisfy condition ($\star$).

From a high-level perspective one can probably characterize the construction $F\mapsto \tilde F$ as a special left biadjoint to the equipment-inclusion from categories to reflections. In Section 4.2 of [2] I consider a variant / special case of this biadjunction of equipments, but in my setting, instead of $\dagger$, the special $1$-cells satisfy the stronger condition of preserving the reflective subcategories, which you also allude to.

[1] Verity, Dominic. Enriched categories, internal categories and change of base. Diss. University of Cambridge, 1992.

[2] Frey, Jonas. "Triposes, q-toposes and toposes." Annals of pure and applied logic 166.2 (2015): 232-259.