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We introduce the algebras satisfying the $(ℬ,n)$ condition. If $\Lambda$, $\Gamma$ are algebras satisfying the $(ℬ,n)$, $(ℰ,m)$ condition, respectively, we give a construction of $(m+n)$-almost split sequences in some subcategories ${(ℬ\otimes ℰ)}^{(i_0,j_0)}$ of $mod(\Lambda \otimes \Gamma )$ by tensor products and mapping cones. Moreover, we prove that the tensor product algebra $\Lambda \otimes \Gamma$ satisfies the $({(ℬ\otimes ℰ)}^{(i_0,j_0)},n+m)$ condition for some integers $i_0$, $j_0$; this construction unifies and extends the work of A. Pasquali (2017), (2019).