Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; \otimes_\Lambda \Gamma} \mathbf{Mod_\Gamma} \xrightarrow{\operatorname{Hom}_\Gamma(\Gamma,\;\cdot\;)} \mathbf{Mod_\Lambda}, $$ and $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; \otimes_\Lambda \Gamma} \mathbf{Mod_\Gamma} \xrightarrow{\;\cdot\;_\Lambda} \mathbf{Mod_\Lambda}, $$ as well as $$ \mathbf{Mod_\Gamma} \xrightarrow{\;\cdot\;_\Lambda} \mathbf{Mod_\Lambda} \xrightarrow{\operatorname{Hom}_\Lambda(\Gamma,\;\cdot\;)} \mathbf{Mod_\Gamma}. $$
Then $\operatorname{Hom}_\Gamma(\Gamma,\;\cdot\;)$ is naturally isomorphic to $\;\cdot\;_\Lambda$ and thus is left-adjoint to $\operatorname{Hom}_\Lambda(\Gamma,\;\cdot\;)$.
Is all this correct?