= Solution
Construct <pointwise limits in a functor category>. For $H:\mathcal J\to[\mathcal D,\mathcal C]$, choose at each object $d$ a <categorical limit> $L(d)$ of $j\mapsto H_j(d)$, with projections $p_j(d)$. For $u:d\to d'$, the family $H_j(u)p_j(d)$ is compatible; define $L(u)$ uniquely by
$$
p_j(d')L(u)=H_j(u)p_j(d).
$$
The <universal property> gives $L(1_d)=1_{L(d)}$ and $L(vu)=L(v)L(u)$, since those equalities hold after every projection. Thus $L$ is a <functor>, and each $p_j:L\Rightarrow H_j$ is a <natural transformation>.
For any <categorical cone> $(M,(a_j))$, the pointwise <categorical limits> give unique maps $a(d):M(d)\to L(d)$. To check naturality, compose $L(u)a(d)$ and $a(d')M(u)$ with every $p_j(d')$; both become $H_j(u)a_j(d)=a_j(d')M(u)$. The projections distinguish arrows into their <categorical limit>, so the two maps agree. Componentwise uniqueness gives uniqueness of the <natural transformation> $a$. This proves that $(L,p_j)$ really is the required <categorical limit>, rather than merely a family of objectwise candidates.
A specified <categorical limit> <categorical cone> in $\mathcal C^{\operatorname{ob}\mathcal D}$ fixes these objectwise vertices and projections. The displayed equation forces every arrow $L(u)$, and the argument forces every <categorical cone> factorization. Hence the <forgetful functor> uniquely lifts that <categorical cone> and is a <limit-creating functor>. The construction only takes small <categorical limits> in $\mathcal C$; it does not require $\mathcal D$ to be small. As usual, the <functor categories> are understood in a universe where their collections of transformations are meaningful.
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