I"ve posted this question https://math.stackexchange.com/questions/4365925/tangent-space-of-g-times-h-m?noredirect=1#comment9122257_4365925 in MSE, but didn't get any answer. My question is the following:

Let $G$ be a Lie group and let $H$ be a Lie subgroup of $H$. Let $M$ be a smooth manifold on which $H$ acts from the left.  

Let's consider the  action of  $H$ on $G \times M$ : $$h((g,m)):= (gh,h^{-1}m), \quad h \in H , g \in G , m \in M, $$ and define the manifold $Z$ to be  the quotient $G  \times_H M .$

If we fix $(g,m) \in G \times M$, what is the induced equivalence relation on $T_gG  \times T_mM$ ?


(My background is not so good in differential geometry, so please be patient with me.)