Let G be a Lie group and $H\subseteq G$ be a closed subgroup. It can be shown that G has a unique differentiable structure such that the inclusion map $H\to G$ is an embedding of manifolds. The functor of points $h_G=\text{Hom}(-,G)$, $h_H=\text{Hom}(-,H)$ of the Lie groups G and H define group objects in the category of manifolds. The inclusion map $H\to G$ gives a natural transformation $h_H\to h_G$ which clearly makes $h_H$ into a subfunctor of $h_G$. Therefore we may take the quotient functor Q which takes each manifold M to the quotient $Q(M):=\text{Hom}(M,G)/\text{Hom}(M,H)$.

Is this functor $Q$ a sheaf on the Grothendieck pretopology where the coverings of an arbitary manifold M are given by the sets of arrows {$\iota_i:U_i \to M $ } such that each $\iota_i$ is a diffeomorphism of $U_i$ onto an open subset of M and $\bigcup \iota_i(U_i)=M$?

(An analogous question is often posed in the context of algebraic groups, however, I am interested in this differentiable setting.)

Let G be a Lie group and $H\subseteq G$ be a closed subgroup. It can be shown that $H$ has a unique differentiable structure such that the inclusion map $H\to G$ is an embedding of manifolds. The functor of points $h_G=\text{Hom}(-,G)$, $h_H=\text{Hom}(-,H)$ of the Lie groups G and H define group objects in the category of manifolds. The inclusion map $H\to G$ gives a natural transformation $h_H\to h_G$ which clearly makes $h_H$ into a subfunctor of $h_G$. Therefore we may take the quotient functor Q which takes each manifold M to the quotient $Q(M):=\text{Hom}(M,G)/\text{Hom}(M,H)$.

Is this functor $Q$ a sheaf on the Grothendieck pretopology where the coverings of an arbitary manifold M are given by the sets of arrows {$\iota_i:U_i \to M $ } such that each $\iota_i$ is a diffeomorphism of $U_i$ onto an open subset of M and $\bigcup \iota_i(U_i)=M$?

(An analogous question is often posed in the context of algebraic groups, however, I am interested in this differentiable setting.)