a pointwise dualizing module $\omega$ and that $M$ is a non-zero finitely generated $R$-module with locally finite ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$. Fact \[fact110612a\] implies that ${C\otimes_{R}M}$ has locally finite injective dimension. Since $M\neq 0$, we have ${C\otimes_{R}M}\neq 0$, by Fact \[fact110527a\](b), so Theorem \[thm110530a\] provides an integer $r = {\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}M})})$, a proper $R$-regular sequence $\underline z = z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$\label{proof110609b1}
0 \to {C\otimes_{R}M} \to Z' \to N' \to 0$$ with $Z' = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$, where $N'$ has locally finite injective dimension and ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,N')}){\leqslant}r$.

As ${C\otimes_{R}M}$ has locally finite injective dimension, we have ${C\otimes_{R}M}\in{{\mathcal{B}}_C}(R)$ by Fact \[disc01xy\](b), so ${\operatorname{Ext}_{R}^{1}(C,{C\otimes_{R}M})}=0$ and ${\operatorname{Hom}_{R}(C,{C\otimes_{R}M})}\cong M$. Thus, an application of ${\operatorname{Hom}_{R}(C,-)}$ to the sequence  yields the next exact sequence: $$\label{proof110609b2}
0 \to M \to {\operatorname{Hom}_{R}(C,Z')} \to {\operatorname{Hom}_{R}(C,N')} \to 0.$$ Since $N'$ has locally finite injective dimension, Fact \[fact110612a\] implies that the $R$-module $N:={\operatorname{Hom}_{R}(C,N')}$ has locally finite ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$. With $Z:={\operatorname{Hom}_{R}(C,Z')}$, it remains to show that $Z \cong \oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}$ and ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})}){\leqslant}r$. The first of these follows from the next sequence of isomorphisms $$\begin{aligned}
Z
&\cong{\operatorname{Hom}_{R}(C,\oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i})}\\
&\cong\oplus_{i=0}^r{\operatorname{Hom}_{R}(C,{\omega\otimes_{R}(R/(\underline z_{[i]})})}^{n_i}\\
&\cong\oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}\end{aligned}$$ where the last isomorphism is from Lemma \[lem110612b\]. To complete the proof, observe that 