Let $\mathbf C$ be a complete category and let $U:\mathbf C\to\mathbf{Set}$ be a representable functor. Show that $U$ preserves limits.
In general representables preserve limits, but the hypothesis of completeness suggests me to prove that $U$ is a right adjoint (and thus preserves limits). Let $U\cong \mathbf C(r,-)$. In order to define a left adjoint $F\dashv U$, I reasoned like this: $\mathbf {Set}(\{*\},U(c))\cong U(c)\cong \mathbf C(r,c)$, so we can set $F(\{*\})=r$; since $F$ is a left adjoint it preserves colimits, and necessarily $F(S)=F({^S\{*\}})={^SF(\{*\})}={^Sr}$, where with $^S-$ I denote the copower (in any category) indexed on $S$. However $\mathbf C$ is complete, not cocomplete, so the existence of copowers is not guaranteed. Am I doing something wrong or should the hypothesis of the exercise be on cocompleteness (to imply that $U$ is a right adjoint)? Thank you in advance
I think you're overthinking this. Rather than proving that $U$ has a left adjoint, it seems the intention is just to prove it preserves limits. The assumption of completeness is not necessary - it was probably included to avoid any ambiguity about the meaning of "preserves limits" in the case when certain limits do not exist.
You're correct that a representable functor to $\mathsf{Set}$ has a left adjoint exactly when all copowers of the representing object exist. This is a non-trivial hypothesis. For example, the inclusion of the category of finite sets into the category of sets is representable (by the singleton set) but does not have a left adjoint.