$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Stated more informally, Proposition asserts that the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is universal among categories $\operatorname{\mathcal{E}}$ which are equipped with a pair of functors $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{E}}\xleftarrow {G} \operatorname{\mathcal{D}}$ and a natural transformation $v: (F \circ \pi _{\operatorname{\mathcal{C}}}) \rightarrow (G \circ \pi _{\operatorname{\mathcal{D}}})$. More precisely, there is a pushout square

\[ \xymatrix@R =50pt@C=50pt{ (\operatorname{\mathcal{C}}\times \{ 0 \} \times \operatorname{\mathcal{D}}) \coprod ( \operatorname{\mathcal{C}}\times \{ 1\} \times \operatorname{\mathcal{D}}) \ar [r] \ar [d] & \operatorname{\mathcal{C}}\times [1] \times \operatorname{\mathcal{D}}\ar [d] \\ ( \operatorname{\mathcal{C}}\times \{ 0\} ) \coprod ( \{ 1\} \times \operatorname{\mathcal{D}}) \ar [r] & \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}} \]

in the (ordinary) category $\operatorname{Cat}$, where the right vertical map encodes the natural transformation $u: \iota _{\operatorname{\mathcal{C}}} \circ \pi _{\operatorname{\mathcal{C}}} \rightarrow \iota _{\operatorname{\mathcal{D}}} \circ \pi _{\operatorname{\mathcal{D}}}$ appearing in the proof of Lemma