Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 22 / 2 / d / Solution

For every $C$, the preceding part identifies the composite $\mathcal{C}(C,{\lim }_J-)$ with the covariant representable functor $[J,\mathcal{C}]({\Delta }_{J}C,-)$. 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. The limit functor preserves limits. The hom-set detection of categorical limits givesIn 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 ${\Delta }_J⊣{\lim }_J$.