  We study the categorical structure of the Grothendieck construction of an
indexed category $\mathcal{L}:\mathcal{C}^{op}\to\mathbf{CAT}$ and characterise
fibred limits, colimits, and monoidal structures. Next, we give sufficient
conditions for the monoidal closure of the total category $\Sigma_\mathcal{C}
\mathcal{L}$ of a Grothendieck construction of an indexed category
$\mathcal{L}:\mathcal{C}^{op}\to\mathbf{CAT}$. Our analysis is a generalization
of G\"odel's Dialectica interpretation, and it relies on a novel notion of
$\Sigma$-tractable monoidal structure. As we will see, $\Sigma$-tractable
coproducts simultaneously generalize cocartesian coclosed structures,
biproducts and extensive coproducts. We analyse when the closed structure is
fibred -- usually it is not.
