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. 

Let $X$ be a vector field on $G$. For each $g\in G,$ the map $g\mapsto xg$ is a diffeomorphism of $G$ so the pushforward $(L_x)_*:T_gG\to T_{xg}G$ is well-defined.  
Now, $(L_x)_*$ maps vectors in $T_gG$ to vectors in $T_{xg}G$. And so if $X$ is a vector field on $G$, then $(L_x)_*X_g$ is $some$ vector $v_{xg}\in T_{xg}G$. 
$\textit{If}\ $  it turns out that $v_{xg}=X_{xg}$ for every $g\in G,$  we say that $X$ is left-invariant. Of course, this does not say that there are any such fields, but this is another matter. 
