I've made the following observation: Let $i : Z \subseteq X$ the inclusion of a closed subset of a topological space. Then the functor

$i_* : Sh(Z) \to \{F \in Sh(X) : supp(F) \subseteq Z\}$

is an equivalence of categories with quasi-inverse $i^{-1}$. Remark that here $i^{-1}$ admits a simple description (or rather realization of its universal property being adjoint to $i_*$), namely $(i^{-1} F)(W) = F(\tilde{W})$, where $\tilde{W}$ is the greatest open subset of $X$ such that $\tilde{W} \cap Z = W$. We have an isomorphism $id \cong i_* i^{-1}$, but we even have an *equality* $id = i^{-1} i_*$ (because $(i^{-1} i_* F)(W) = (i_* F)(\tilde{W}) = F(\tilde{W} \cap Z) = F(W)$, compatible with restrictions).

More generally, we could define an equimorphism (mix of equivalence and isomorphism) to be a functor $F : C \to D$ such that there exists a functor $G : D \to C$ such that $1_C = GF$ and $1_D \cong FG$.

Do equimorphisms appear in other contexts as well? Is there some literature about them? What is the intuition behined them, how can I see directly that a known equivalence is actually an equimorphism? Is it possible to turn an equivalence (perhaps even by changing the categories by pure set-thoretically means, leaving the structure the same) into an equimorphism?