This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes
For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has tangent map given by $(A,X)\mapsto(A\circ B, A\circ X)$, where $(A,X)\in GL(V)\times L(V,V)\cong T(GL(V))$
I feel that this must be a typo, since surely he is just talking about left multiplication when the group is $GL(V)$ and $A\mapsto A\circ B$ doesn't make sense to me since there is no mention of $B$ anywhere. Surely he means $B\mapsto A\circ B$?
Also, should the tangent map be given by $(B,X)\mapsto (A\circ B, A\circ X)$?
Yes, I came across the same question. It would make sense to be $B \mapsto A \circ B$. From this as $L_A$ is linear we would get $$TL_A|_{T_B(GL(V))} : T_B(GL(V)) \simeq \{B\} \times L(V,V) \to T_{L_A(B)}(GL(V)) \simeq \{A \circ B\} \times L(V,V)$$ given by
$$\begin{align}TL_A|_{T_B(GL(V))} (B,X) &= (A \circ B, DL_A(X)) \\&= (A \circ B, L_A(X))\\&= (A \circ B, A \circ X) \end{align}$$