Additive cocontinuous module functors admit right adjoints

By [F1] there is a natural isomorphism $\sigma :T_M\Rightarrow F$ with inverse ${\sigma }^{-1}:F\Rightarrow T_M$, and $M$ is a $(B,A)$-bimodule. By [F2] there are a unit $\eta :1\Rightarrow G{T}_M$ and a counit $\epsilon :T_{M}G\Rightarrow 1$ satisfying the triangle identities of [F3] for the adjunction $T_M⊣G$.

Transfer: put ${\eta }^{\mathrm{\prime }}:=(G\sigma )\circ \eta :1\Rightarrow GF$ and ${\epsilon }^{\mathrm{\prime }}:=\epsilon \circ ({\sigma }^{-1}G):FG\Rightarrow 1$, using the whiskerings of [F4]; these are natural transformations by [F4] and [F5]. They satisfy the triangle identities of [F3]: at $X$, using naturality of ${\sigma }^{-1}$ at $G({\sigma }_X)$, naturality of $\sigma$ at ${\eta }_X$ and naturality of $\epsilon$ at ${\sigma }_X$, one computes ${\epsilon }_{F(X)}^{\mathrm{\prime }}\circ F({\eta }_X^{\mathrm{\prime }})={\epsilon }_{F(X)}\circ ({\sigma }^{-1}{)}_{G(F(X))}\circ F(G({\sigma }_X))\circ F({\eta }_X)={\epsilon }_{F(X)}\circ T_M(G({\sigma }_X))\circ T_M({\eta }_X)\circ {\sigma }_X^{-1}={\sigma }_X\circ {\epsilon }_{T_M(X)}\circ T_M({\eta }_X)\circ {\sigma }_X^{-1}={\sigma }_X\circ {\sigma }_X^{-1}=1_{F(X)}$; and at $Y$ one computes $G({\epsilon }_Y^{\mathrm{\prime }})\circ {\eta }_{G(Y)}^{\mathrm{\prime }}=G({\epsilon }_Y)\circ G(({\sigma }^{-1}{)}_{G(Y)})\circ G({\sigma }_{G(Y)})\circ {\eta }_{G(Y)}=G({\epsilon }_Y)\circ {\eta }_{G(Y)}=1_{G(Y)}$, where the middle step cancels the inverse components of [F5]. Hence $F⊣G$ in the sense of [F3].

By step 2.1 the functor $F$ admits $G={\mathrm{Hom}}_B(M,-)$ as right adjoint, so it is a left adjoint and [F6] applies; in particular it preserves every colimit that exists, consistently with its assumed cocontinuity. The transfer used only the displayed units, counits and inverse components, so no commutativity and no choice are involved.