# Tensor-hom adjunction in a general closed monoidal category

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)?

• 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? – Victor Apr 29 at 2:48