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

Conversely, assume that every hom-set functor $\mathcal{C}(C,F-)$ preserves existing categorical limits. For a categorical limit cone ${\ell }_i:L\to E_i$ in $\mathcal{B}$, a categorical cone $q_i:C\to F{E}_i$ determines an element of ${\lim }_i\mathcal{C}(C,F{E}_i)$. The assumed bijectiongives one and only one such $u$ for every $C$ and every categorical cone. That is the universal property of the image cone itself. Therefore the hom-set tests detect limit preservation:This proof works for all diagram sizes for which the relevant categorical limits and compatible-family sets are under consideration.