= Solution
Fix the chosen <categorical limit> object $LA$ and its <categorical cone> $p_{A,j}:LA\to(DA)_j$. For $a:A\to A'$, the family
$$
LA\xrightarrow{p_{A,j}}(DA)_j\xrightarrow{(Da)_j}(DA')_j
$$
is a <categorical cone>, by <naturality> of $Da$. The <universal property> of $LA'$ supplies a unique <morphism>
$$
\boxed{La:LA\to LA',\qquad p_{A',j}La=(Da)_jp_{A,j}\text{ for every }j.}
$$
The <identity morphism> of $LA$ satisfies the equations for $a=1_A$, so $L1_A=1_{LA}$. For composable $a,a'$, the equations for $La'\,La$ agree with those for $L(a'a)$; uniqueness gives $L(a'a)=La'\,La$. Hence \b[the chosen limits define a functor] $L:\mathcal A\to\mathcal C$. This argument also handles an empty indexing <category>, when each chosen <categorical limit> is a <terminal object>.
Equivalently, $LA$ represents the <categorical cone> <functor> $C\mapsto\operatorname{Nat}(\Delta_JC,DA)$, and <functoriality of chosen representations> supplies the same $L$.
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