The Stacks project

Lemma 24.17.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ and $\mathcal{B}$ be a sheaves of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{N}$ be a right differential graded $\mathcal{B}$-module. There is a $1$-to-$1$ correspondence between $(\mathcal{A}, \mathcal{B})$-bimodule structures on $\mathcal{N}$ compatible with the given differential graded $\mathcal{B}$-module structure and homomorphisms

\[ \mathcal{A} \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^{dg}_\mathcal {B}(\mathcal{N}, \mathcal{N}) \]

of differential graded $\mathcal{O}$-algebras.

Proof. Omitted. $\square$

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