$\textbf{Problem.}$ Let $G$ be Lie group. Let $F:G\times G\rightarrow G$ denote the multiplication map. Identify the space $T_{(e,e)}(G\times G)$ with $T_{e}G\oplus T_{e}G$ by $$v\in T_{(e,e)}(G\times G)\mapsto\bigg(d\pi_{1}(v),d\pi_{2}(v)\bigg)$$ where $\pi_{1}$ and $\pi_{2}$ are projections of $G\times G$ to the first and the second components, respectively. Show that $dF_{e}:T_{e}G\oplus T_{e}G\rightarrow T_{e}G$ is given by $dF(X,Y)=X+Y$.
I was little bit confused about interpreting the last identity.
If I plug in a function $f$ defined on $G$ on the both sides, the LHS becomes $dF(X,Y)(f)=(X,Y)(f\circ F)$ and the RHS becomes $(X+Y)(f)$. But it looks like then the LHS became a function on $G\times G$ and the RHS became a function on $G$. It doesn't look right to me so please let me know how to understand this.
I believe that the shorthand $\Bbb d F (X, Y) = X + Y$ means the following (it is important to apply it at points, to make thnigs clear). I shall use lower indices for applications at points.
If $F(g,h) = gh$, then $\Bbb d F : T_g G \oplus T_h G \to T_{gh} G$ is defined as follows:
$$\Bbb d _{(g,h)} F \ (X_g, Y_h) = \Bbb d _g (R_h) (X_g) + \Bbb d _h (L_g) (Y_h) .$$
Applying to some $f : G \times G \to G$, the right-hand side becomes $(f \circ R_h) (X_g) + (f \circ L_h) (Y_h)$, which is now fine.