Lemma 4.28.2. The horizontal and vertical compositions have the following properties

$\circ $ and $\star $ are associative,

the identity transformations $\text{id}_ F$ are units for $\circ $,

the identity transformations of the identity functors $\text{id}_{\text{id}_\mathcal {A}}$ are units for $\star $ and $\circ $, and

given a diagram

\[ \xymatrix{ \mathcal{A} \rruppertwocell ^ F{t} \ar[rr]_(.3){F'} \rrlowertwocell _{F''}{t'} & & \mathcal{B} \rruppertwocell ^ G{s} \ar[rr]_(.3){G'} \rrlowertwocell _{G''}{s'} & & \mathcal{C} } \]we have $ (s' \circ s) \star (t' \circ t) = (s' \star t') \circ (s \star t)$.

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