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Let $(M,p)\xrightarrow{\ \ F\ \ }(S,s)\xleftarrow{\ \ G\ \ }(N,q)$ be smooth maps of pointed differentiable manifolds. If $F$ and $G$ are transversal then the fibered product $(M,p)\times_{(S,s)}(N,q)$ exists, the underlying pointed topological space is given by the fibered product $(M\times_SN,(p,q))$ and the tangent space is given by the fibered product $$T_{(p,q)}( (M,p)\times_{(S,s)}(N,q))=T_p(M)\times_{T_s(S)}T_q(N)\,.$$In other words, fibered products of transversal morphisms of pointed differentiable manifolds exists and are preserved both by the forgetful functor to the category of pointed topological spaces and by the tangent space functor to the category of $\mathbb{K}$-vector spaces. A reference for this is Theorem 5.47 in Manifolds, Sheaves and Cohomology by Torsten Wedhorn. I know that not all fibered products exist in the category of (pointed) manifolds, see for example here. However, my question is the following:

Do the forgetful functor $\mathbf{Man}_\ast\longrightarrow\mathbf{Top}_\ast$ and the tangent space functor $\mathbf{Man}_\ast\longrightarrow\mathbf{Vect}_\mathbb{K}$ commute with the formation of fibered products in the categorical sense? I.e. are universal cones sent to universal cones whenever they exist?

Is there anything unclear about the question or has nobody ever thought about it?

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A functor which preserves fiber products also preserves monomorphisms. But there are injective smooth maps whose differential is not injective, for example $\mathbb{R} \to \mathbb{R}$, $x \mapsto x^3$ at the origin.

Thus, $T$ does not preserve fiber products. I am not sure about the forgetful functor to spaces, though.

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