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First, I apologize if this question is poorly-worded or otherwise vague, I'll try to be as clear as possible.

If $F:N\rightarrow M$ is a smooth map between smooth manifolds $N$ and $M$, then at each point $p \in N$ the map $F$ induces the derivation $F_{*p}:T_pN \rightarrow T_{F(p)}M$ between tangent spaces, called the differential, that is determined by $F_{*p}(X_p)f = X_p(f \circ F)$ for all smooth real-valued functions $f$ on $M$.

To me, this seems like a covariant functor from the category of smooth manifolds to the category (?) of tangent spaces. My understanding though if it is to be a functor it must also assign, for example, a manifold $M$ to a tangent space $T_pM$. Are there additional aspects of defining the differential that would facilitate this?

Is there a way, perhaps, that this can be achieved with the inclusion maps $i_N$ and $i_M$ of $N$ and $M$ into $T_pN$ and $T_pN$ since the differential, satisfies $F_{*p} \circ i_N = i_N \circ F$?

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