Fulton and Harris, Representation Theory, Section 8.1 (pages 104 - 107 in my copy) is concerned with showing that group homomorphisms $\rho : G \to H$, where $G$ is connected, are completely determined by $(d\rho)_e : T_e G \to T_e H$. This is used to motivate the definition of a Lie algebra, which comes later on.
They show that $\rho$ being a group homomorphism implies a string of statements. Up to the statement that
$$\require{AMScd} \begin{CD} T_e G @>{d\rho_e}>> T_e H\\ @V{\operatorname{Ad}(g)}VV @VV{\operatorname{Ad}(\rho(g))}V \\ T_e G @>>{d\rho_e}> T_e H \end{CD}$$
commutes, I'm fully on board. But then it's claimed that this implies that
$$\require{AMScd} \begin{CD} T_e G @>{d\rho_e}>> T_e H\\ @V{\operatorname{ad}(v)}VV @VV{\operatorname{ad}(d\rho_e(v))}V \\ T_e G @>>{d\rho_e}> T_e H \end{CD}$$
commutes as well, with no explanation given so far as I can see. How do you get from one to the next? It seems as though the idea would be to show that every $\operatorname{ad}(v)$ comes from some $\operatorname{Ad}(g)$, but there doesn't seem to be sufficient machinery to do this -- both $\operatorname{Ad}$ and $\operatorname{ad}$ have just been defined on this page!