making sense of hom-set between chain complex and abelian group

Question

Let $\textbf C$ be a chain complex of right $R$-modules, let $A$ be a left $R$-module and let $B$ be an abelian group.

How to make sense of the $\mathrm{Hom}(\textbf {C}\otimes_R A,B)$ and $\mathrm{Hom}_R(\textbf {C},\mathrm{Hom}(A,B))$?

Answer 1

Usually, if $\mathbf C$ denote a complex $$\cdots\to C_n\xrightarrow{\gamma_n}C_{n+1}\to\cdots$$ then $\DeclareMathOperator\Hom{Hom}\Hom(\mathbf C\otimes_RA,B)$ denote the complex $$\cdots\leftarrow\Hom(C_n\otimes_RA,B)\xleftarrow{\Hom(\gamma_n\otimes_R1_A,1_B)}\Hom(C_{n+1}\otimes_RA,B)\leftarrow\cdots$$ while $\Hom(\mathbf C,\Hom(A,B))$ denote the complex: $$\cdots\leftarrow\Hom_R(C_n,\Hom(A,B))\xleftarrow{\Hom_R(\gamma_n,\Hom(1_A,1_B))}\Hom_R(C_{n+1},\Hom(A,B))\leftarrow\cdots$$