Remark 6.1.5.8. In the situation of Construction 6.1.5.7, we can identify $\epsilon $ and $\epsilon '$ with $2$-morphisms
\[ \epsilon ^{\operatorname{c}}: \operatorname{id}_{D^{\operatorname{c}}} \Rightarrow f^{\operatorname{c}} \circ g^{\operatorname{c}} \quad \quad \epsilon '^{\operatorname{c}}: \operatorname{id}_{ E^{\operatorname{c}}} \Rightarrow f'^{\operatorname{c}} \circ g'^{\operatorname{c}} \]
in the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$ (Construction 2.2.3.4). The contraction $c( \epsilon , \epsilon ' )$ can then be described as the conjugate of the $2$-morphism $c( \epsilon '^{\operatorname{c}}, \epsilon ^{\operatorname{c}} )$ obtained by applying Construction 6.1.5.1 to the $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$.