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 P: 150 In my notes, the following two functions are defined: Suppose $M^m$ and $N^n$ are smooth manifolds, $F:M \to N$ is smooth and $p \in M$. We define: $$F^*:C^\infty (F(p)) \to C^\infty (p)\ ,\ F^*(f) = f \circ F$$ $$F_{*p}: T_pM \to T_{F(p)}N\ ,\ [F_{*p}(X)](f) = X(F^*f) = X(f \circ F)$$ I understand the first function, $F^*$; it maps $f$, a function on $C^\infty(F(p))$, to $f \circ F$, a function on $C^\infty(p)$. However, I don't understand the second one, $F_{*p}$. Since $X(f) \in T_pM$, it follows that $f \in C^\infty (p)$. But then how is $$[F_{*p}(X)](f) = X(F^*f)$$ defined? After all, in the definition of $F_{*p}(X)$, $f$ is a function on $C^\infty (p)$, not $C^\infty(F(p))$, so how can we evaluate $F^*f$?