This question arises in the context of Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings.
Let $(R,m)$ be a local complete Cohen-Macaulay ring of dimension $d$. Denote by $H_m^d(-),\omega_R$ the local cohomology functor at dimension $d$ and the canonical module of $R$ reprectively. $T^0(-)=\operatorname{Hom}_R(H_m^d(-),\omega_R)$ is a contravariant left-exact functor that takes direct sums to direct products. Then the authors say that as a consequence, there exists some $R$-module $C$ such that $T^0(-) = \operatorname{Hom}(-,C)$. Could somebody please explain why this last statement is true?
PS: The authors reference theorem 3.36 in the new edition of Rotman's Homological Algebra, however i only have access to the free online version of the first edition.
It's just a direct application of the Eilenberg-Watts theorem [1], and a free version of Watts' paper is available. Basically, the functor $\mathrm{Hom}(-,C)$ is a left-exact contravariant functor that has this property. The Eilenberg-Watts theorem says the converse is true: any left exact additive contravariant functor taking direct sums to direct products is representable, and there are versions for the tensor product, etc. The proof itself is elementary, short, and fun to read!
[1] Watts, Charles E. "Intrinsic characterizations of some additive functors." Proceedings of the American Mathematical Society 11.1 (1960): 5-8.