Definition of quotient manifolds, and $\Gamma \backslash \mathscr H$ as a quotient manifold I have just encountered some subtlety with quotient manifolds and now I don't think I understand some things as well as I thought I did.  
Let $X$ be a real or complex analytic manifold, and $\sim$ an equivalence relation on $X$.  Let $p: X \rightarrow X/\sim$ be the quotient map.  Here are two possible definitions of quotient manifold.


*

*A manifold structure on $X/\sim$, such that $g \mapsto g \circ p$ defines a bijection


$$\operatorname{Hom}(X/\sim, Y) \rightarrow \operatorname{Hom}(X,Y)$$
where the morphisms are taken in the appropriate category (either real or complex manifolds).


*

*A manifold structure on $X/\sim$ such that $p$ is a submersion.


Are these definitions equivalent?  I believe that the second definition implies the first.  
Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$.  It is possible to put a complex manifold structure on the quotient space $\Gamma \backslash \mathscr H$, such that the map $p: \mathscr H \rightarrow \Gamma \backslash \mathscr H$ is (complex) analytic.  However, this map is not a submersion.  If $\tau \in \mathscr H$ is an elliptic point, and $h \geq 2$ is the order of the image of the stabilizer of $\tau$ in $\operatorname{PSL}_2(\mathbb{Z})$, then locally near $\tau$, $p$ looks like the map $z \mapsto z^h$ near $0$, whose derivative vanishes.  
In what sense, if any, can $\Gamma \backslash \mathscr H$ be thought of as a quotient of manifolds?
 A: As is often the case, a slight recasting of the question may produce a much-easier-to-answer question which suffices for the actual purposes at hand.
As in the comment and in the original question, already for quotients of the upper half-plane, and all the more so for quotients of classical domains $G/K$ for higher-$\mathbb R$-rank Lie groups and maximal compacts $K$, the natural quotient structure at least appears to give the quotient singularities.
It is of course interesting to try to understand various desingularization possibilities... but this is a difficult issue, I think.
In contrast, for elementary reasons $\Gamma\backslash G$ is always a smooth manifold, so $C^\infty_c(\Gamma\backslash G)$ immediately makes sense. And, then, instead of trying to define $C^\infty_c(\Gamma\backslash G/K)$ directly in terms of some (not reliably existent!) smooth structure on $\Gamma\backslash G/K$, for many purposes it suffices to consider the right $K$-fixed functions $C^\infty_c(\Gamma\backslash G)^K$ in $C^\infty(\Gamma\backslash G)$...
