Let$F:M \to N$ be a smooth map of manifolds. The pointwise pushforward is a linear map $$ F_{*,p} : T_p M \to T_{F(p)} N; \qquad v \mapsto F_{*,p}v: C^\infty(N) \ni \phi \mapsto v (\phi \circ F) \in \mathbb R $$ and the pullback is its dual map $$ F^*_p: T_{F(p)}^*N \to T_p^* M; \qquad f \mapsto f \circ F_{*,p} $$
The question is: when one extends these maps to fields (even general tensor fields) on the whole manifold, then what should domain and the codomain of these maps be ? e.g. for the pushforward is it $$ F_*: TM \to TN; \qquad \text{or} \qquad F_*:\Gamma(TM) \to \Gamma(TN) \quad? $$ in the first case it is a map of smooth manifolds, in the other it is an $\mathbb R$-linear map of vector spacees (or even a module homomorphism, right ?). Similarly for the pullback.