= Solution
The representing object in the <functor category> $[J,\mathcal C]$ is the constant <diagram in a category> $\Delta_JC$. If $p_j:\lim_jE_j\to E_j$ is the chosen <categorical limit> cone, the <natural bijection> is
$$
\boxed{\mathcal C(C,\lim_jE_j)\cong[J,\mathcal C](\Delta_JC,E),
\qquad f\longmapsto(p_jf)_j.}
$$
A <natural transformation> from the constant <diagram in a category> is precisely a <categorical cone> with vertex $C$, and its inverse is the unique mediating <morphism> from the <universal property> of the <categorical limit>. For $\alpha:E\Rightarrow E'$, the defining equations for $\lim\alpha$ show that postcomposition by $\lim\alpha$ corresponds to postcomposition by $\alpha$ on the right. This establishes <naturality> in $E$ and proves \b[representability by the constant diagram].
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