Notation 2.2.6.7. Let $\operatorname{\mathcal{C}}$ be a $2$-category. A twisting cochain for $\operatorname{\mathcal{C}}$ is a datum which assigns, to every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, a $1$-morphism $(g \circ ' f): X \rightarrow Z$ and an invertible $2$-morphism $\mu _{g,f}: g \circ ' f \xRightarrow {\sim } g \circ f$. In this case, we will (slightly) abuse notation by identifying the twisting cochain with the collection of $2$-morphisms $\{ \mu _{g,f} \} $.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$