Endocoherent complexes of modules

Let $R$ be an associative ring with identity. A complex $C$ of $R$-modules is called <span class="italic">coherent</span> if it is finitely generated and every finitely generated subcomplex of $C$ is finitely presented. Suppose that $C = (C_i, d_i)_{i\in \mathbb{Z}}$ is a complex of right $R$-modules and $S$ is the endomorphism ring of $C$. There is a natural action of $S$ on each term $C_i$ so that $C = (C_i, d_i)_{i\in \mathbb{Z}}$ becomes a complex of left $S$-modules. It is proved that $S$ is a left coherent ring if and only if every complex $P \in$ $\operatorname{pres}C$ has an $\operatorname{add}C$-preenvelope.  Moreover, if $C$ is finitely presented, then it is coherent as a complex of left $S$-modules if and only if every finitely presented complex of right $R$-modules has an $\operatorname{add}C$-preenvelope.