Let $(X,\mathcal O_X)$ be a ringed space (if necessary, assume that it is a scheme with suitable hypotheses). Given two complexes of sheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules, one has the *complex of morphisms* \begin{equation} \operatorname{Hom}^\bullet(\mathcal F, \mathcal G),\end{equation}
and moreover the *complex of sheaves of morphisms* \begin{equation} \mathcal H om^\bullet (\mathcal F, \mathcal G), \end{equation}
which is defined on any open $U \subseteq X$ by $\mathcal H om^\bullet (\mathcal F, \mathcal G)(U) = \operatorname{Hom}^\bullet (\mathcal F_{|U}, \mathcal G_{|U})$.

We give the following definition:

A complex of sheaves of $\mathcal O_X$-modules $\mathcal I$ is called

K-injective(orhoinjective) if for any exact (acyclic) complex of sheaves of $\mathcal O_X$-modules $\mathcal A$, the complex $\operatorname{Hom}^\bullet(\mathcal A, \mathcal I)$ is acyclic.

This definition should be precisely the same as the one I can found in the literature, for example here. My question is if the following characterisation holds:

A complex of $\mathcal O_X$-modules $\mathcal I$ is K-injective if and only if for any acyclic complex of $\mathcal O_X$-modules $\mathcal A$, the complex of sheaves $\mathcal H om^{\bullet}(\mathcal A, \mathcal I)$ is acyclic.

One implication is trivial because $\operatorname{Hom}^\bullet(\mathcal F, \mathcal G) = \mathcal H om^\bullet (\mathcal F, \mathcal G)(X)$ by definition, but I'm confused about the other one.