0N7W
Lemma 3 . Suppose π ~ \tilde{\mathcal{C}} ( π , β , i ) (\mathcal{C},\otimes,i) π \mathcal{C} β : π β G π β π \otimes\colon\mathcal{C}\otimes_{\rm G}\mathcal{C}\to\mathcal{C} i : β β π i\colon\mathcal{I}\to\mathcal{C}
(1)
π β G π β G π {\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}\otimes_{\rm G}\mathcal{C}} π β G π {\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}} π β G π {\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}} π {\lx@inpgf@ignorespaces\mathcal{C}} β β G π \scriptstyle{\lx@inpgf@ignorespaces\otimes\otimes_{\rm G}\mathcal{C}} π β G β \scriptstyle{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\otimes} β \scriptstyle{\lx@inpgf@ignorespaces\otimes} β \scriptstyle{\lx@inpgf@ignorespaces\otimes}
(2)
β β G π {\lx@inpgf@ignorespaces\mathcal{I}\otimes_{\rm G}\mathcal{C}} π β G π {\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}} π {\lx@inpgf@ignorespaces\mathcal{C}} i β G π \scriptstyle{\lx@inpgf@ignorespaces i\otimes_{\rm G}\mathcal{C}} β
\scriptstyle{\lx@inpgf@ignorespaces\cong} β \scriptstyle{\lx@inpgf@ignorespaces\otimes} βββ π β G β {\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{I}} π β G π {\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}\mathcal{C}} π {\lx@inpgf@ignorespaces\mathcal{C}} π β G i \scriptstyle{\lx@inpgf@ignorespaces\mathcal{C}\otimes_{\rm G}i} β
\scriptstyle{\lx@inpgf@ignorespaces\cong} β \scriptstyle{\lx@inpgf@ignorespaces\otimes}
Conversely, for any ( π , β , i ) (\mathcal{C},\otimes,i) π ~ \tilde{\mathcal{C}} ( π , β , i ) (\mathcal{C},\otimes,i)
0N7X
Proof. This is a straightforward consequence of the definition of semistrict 3-categories as categories enriched over 2 β π’πΊπ 2\mathsf{Cat}