= Solution
For every $C$, the preceding part identifies the composite $\mathcal C(C,\lim_J-)$ with the covariant <representable functor> $[J,\mathcal C](\Delta_JC,-)$. A covariant <representable functor> preserves every existing <categorical limit>, since a <morphism> into a <categorical limit> is the same as a compatible family of <morphisms> into the diagram objects. \b[The limit functor preserves limits.] The <hom-set detection of categorical limits> gives
$$
\boxed{\lim_J:[J,\mathcal C]\to\mathcal C\text{ preserves categorical limits}.}
$$
In particular, it preserves all small <categorical limits>. They exist in $[J,\mathcal C]$: compute each one at every $j$ using completeness of $\mathcal C$, and use the uniqueness of the pointwise mediating <morphisms> to obtain its <functor> structure and universal <natural transformations>. Another expression of the same result is the <adjunction> $\Delta_J\dashv\lim_J$.
Back to article page
