Extension category algebras and LHS--spectral sequences

Let $\mathcal{C}$ be a small category, $\mathfrak{A}$ be a precosheaf of unital $k$-algebras on $\mathcal{C}$ and $\mathfrak{M}$ be an $\mathfrak{A}$-bimodule. We introduce two new notions, namely, the Grothendieck construction $G{r}_{\mathcal{C}}(\mathfrak{A},\mathfrak{M})$ of $\mathfrak{A}$ and $\mathfrak{M}$, as well as the extension category algebra $\mathfrak{A}⋉\mathfrak{M}$. The extension category algebra contains the trivial extension algebra and the skew category algebra as special cases. If $\mathcal{C}$ is object-finite, we prove that the category of modules of $G{r}_{\mathcal{C}}(\mathfrak{A},\mathfrak{M})$ is equivalent to the category of modules over $\mathfrak{A}⋉\mathfrak{M}$. Finally, we obtain two LHS-spectral sequences about $G{r}_{\mathcal{C}}(\mathfrak{A},\mathfrak{N})$ for a right $\mathfrak{A}$-module $\mathfrak{N}$.