Suppose $M$ and $N$ are smooth manifolds. Then, the tangent space $T_{(m,n)}(M\times N)$ decompose as direct sum $T_mM\oplus T_nN$. This has been discussed in many books.
I do not see any reference for Tangent space of pullback manifold.
Suppose $f:M\rightarrow P$ and $g:N\rightarrow P$ are such that they intersect transversally; you can also assume that one of them is a surjective submersion if you want. Then, we know that there is a nice smooth structure on the pullback $M\times_PN$.
Do we have a nice description of $T_{(m,n)}(M\times_P N)$ in terms of $T_mM$ and $T_nN$? I remember proving some time back that $$T_{(m,n)}(M\times_PN)\cong \{(v,w)\in T_mM\times T_nN|f_{*,m}(v)=g_{*,n}(w)\}.$$
But I could not find that notes, I could not prove it quickly now.
Is there any quick proof for this result? Is it mentioned in any book?