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Let $(C,\otimes,1)$ be a closed (not necessarily symmetric) monoidal category with all finite limits and colimits and with the internal hom functor $[b,-]$ right adjoint to $(-)\otimes b$, for any $b\in C$. Let $r$ and $s$ be monoids in $C$ and $x$ be an $(r,s)$-bimodule. Does one always have an adjunction $$ (-)\otimes_r x\colon RMod_r\rightleftarrows RMod_s\colon [x,-]_s, $$ where $(-)\otimes_r x$ is the coequalizer of $(-)\otimes r\otimes x \rightrightarrows (-)\otimes x$ and $[x,-]_s$ is the equalizer of $[x,-]\rightrightarrows [x\otimes s,-]$ (the arrows being naturally defined)?

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  • $\begingroup$ I think I got it. One should in addition require that $Hom_C(−,𝑏)$ converts colimits to limits and $Hom_𝐶(𝑏,−)$ preserves limits. Then one should use the Fubini theorem for limits in sets. Maybe though this statement already appeared in any book or paper? $\endgroup$ – Victor Apr 29 at 2:48

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