Suppose we have three smooth manifolds $M_1$, $M_2$ and $N$ and two smooth maps $f_1:M_1 \rightarrow N$ and $f_2:M_2 \rightarrow N$. Than an important and central construction in differential topology is the $transversal$ $pullback$
$$M_1 \times_{f_1Nf_2} M_2 = \lbrace\left(x_1,x_2 \right) \in M_1 \times M_2 |f_1(x_1)=f_2(x_2) \rbrace$$
A proof that it is a manifold goes like:
$M_1 \times_{f_1Nf_2} M_2 = (f_1 \times f_2 )^{−1}(\Delta)$, where $f_1 \times f_2 : M_1 \times M_2 \rightarrow N \times N$ and where $\Delta$ is the diagonal of $N \times N$ , and $f_1 \times f_2$ is transversal to $\Delta$ if and only if $f_1$ and $f_2$ are transversal.
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Now the question is, can we extend this to multiple transversal pullbacks? For example a "three times pullback":
$M_1 \times_{f_1Nf_2} M_2 \times_{f_2Nf_3} M_3 = \lbrace \left(x_1,x_2,x_3 \right) \in M_1 \times M_2 \times M_3 |f_1(x_1)=f_2(x_2); f_2(x_2)=f_3(x_3) \rbrace$
is this well defined as a smooth manifold and if yes how is it proofed?
And is there another generalization to the $n$-times transversal pullback?
