Small question regarding left invariant vectorfields Usually, one writes out the left invariance of vector fields $X$ on a Lie group $G$ as $$(L_x)_*X=X$$for every $x$. 
However, I have trouble understanding this equality, since both sides does not map to the same domain. Evaluating the right hand side in $y$ gives an element in $T_yG$. Evaluating the left hand side in $y$ yields an element in $T_{xy}G$. Is there some canonical identification between $T_{xy}G$ and $T_yG$ going on, or did I make a mistake somewhere? Or, are we actually evaluating the right hand side in $xy$ instead of $y$?
Thank you. My apologies if this is duplicate. 
I already found this one: Definition of a left-invariant vector field, 
but I do not fully understand the answer. 

The derivative of the left multiplication map, $L_g(x)=gx$ gives an isomorphism of the tangent spaces.  See this.
