contravariant hom-functor dualizes small colimits to limits

The contravariant hom-functor dualizes small limits to colimits, i.e.

\begin{align*} \mathrm{Hom}_{\mathcal{c}}(\mathrm{colim}_{\mathcal{I}} A_i, B) \cong \mathrm{lim}_{\mathcal{I}} \mathrm{Hom}_{\mathcal{C}}(A_i,B_i) \end{align*}

are unique up to unique isomorphism