An epimorphism $\mu:\A\to\B$ of local Weil algebras induces the functor $T^\mu$ from the category of fibered manifolds to itself which assigns to a fibered manifold $p:M\to N$ the fibered product $p^\mu:T^{\bf A}N\times_{T^{\bf B}N}T^{\bf B}M\to T^{\bf A}N$. In this paper we show that the manifold $T^{\bf A}N\times_{T^{\bf B}N}T^{\bf B}M$ can be naturally endowed with a structure of an $\A$-smooth manifold modelled on the $\A$-mod\-ule $\L=\A^n\oplus\B^m$, where $n=\dim N$, $n+m=\dim M$. We extend the functor $T^\mu$ to the category of foliated manifolds $(M,{\cal F})$. Then we study $\A$-smooth manifolds $M^\L$ whose foliated structure is locally equivalent to that of $T^{\bf A}N\times_{T^{\bf B}N}T^{\bf B}M$. For such manifolds $M^\L$ we construct bigraduated cohomology groups which are similar to the bigraduated cohomology groups of foliated manifolds and generalize the bigraduated cohomology groups of $\A$-smooth manifolds modelled on $\A$-mod\-ules of the type $\A^n$. As an application, we express the obstructions for existence of an $\A$-smooth linear connection on $M^\L$ in terms of the introduced cohomology groups.
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