Does "inverting the corresponding morphisms" in a functor have a name? Let $F:C\to D$ be a functor between categories $C$ and $D$. Assume that for every morphism $f:X\to Y$ in $C$, the corresponding morphism $F(f):F(X)\to F(Y)$ is invertible. Let $G(f)$ be the inverse of $F(f)$ and $G(X):=F(X)$. Then $G$ is a (contravariant) functor. The question is, does $G$ have a name? (it is not the inverse of $F$ in the language of category theory).
In addition, if $G$ is a contravariant functor such that $G(X)=F(X)$ and $G(f)\circ F(f)=id$, does $G$ have a name? (it is not a left-inverse for $F$ in the language of category theory).
 A: There is a simple construction explaining what kind of inverses $F$ and $G$ are.
Take $C^{\to}$, the category of arrows of $C$. Then there exist the two functors:
$$
C^{\to}\xrightarrow{\text{dom}_C} C
$$
$$
C^{\to}\xrightarrow{\text{cod}_C} C
$$
and a natural transformation $\text{arr}_C\colon\text{dom}_C\to\text{cod}_C$, such that $\text{arr}_C(f)=f$. This data induce a natural transformation $F_*(\text{arr}_C)\colon F_*(\text{dom}_C)\to F_*(\text{cod}_C)$, which sends $f$ to $F(f)$ and thus can be identified with the functor $F$. If $F$ takes its values in the groupoid of isomorphisms of $D$, then $F_*(\text{arr}_C)$ is a natural isomorphism and its inverse is $G$, which may be identified with the natural transformation $G_*(\text{arr}_C)\colon F_*(\text{cod}_C)\to F_*(\text{dom}_C)$, which sends $f$ to $G(f)$. So $F$ and $G$ are obviously not mutually inverse in $\mathbf{Cat}$, but they induce mutually inverse natural transformations in the category of functors from $C^{\to}$ to $D$.
Same reasoning for the left/right inverses of $F$.
