If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that $\mathrm{Hom}_R(F(A),B) \cong \mathrm{Hom}_R(A,M\otimes_R B)$?
2026-03-27 13:12:31.1774617151
Adjoints to cofree modules tensor?
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$M \otimes_R -$ is almost never continuous and therefore has no left adjoint.