 Solution

Regard the bifunctor as $D^{(2)}:K\to [J,\mathcal{C}]$. Its pointwise categorical limit $E={\lim }_k{D}^{(2)}k$ satisfieswith its action on morphisms obtained by the construction in part (a). Applying the limit-preservation result from part (d) givesThe comparison isomorphism is canonical relative to the chosen categorical limit cones: it is the unique morphism identifying all the projections to $D(j,k)$. This proves commutation of iterated categorical limits, including an empty $J$ or $K$. Equivalently, both iterated constructions have the universal property of the categorical limit of $D$ on $J\times K$.