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

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

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

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

4. 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 $(s' \circ s) \star (t' \circ t) = (s' \star t') \circ (s \star t)$.

Proof. The last statement turns using our previous notation into the following equation

$s'_{F''} \circ {}_{G'}t' \circ s_{F'} \circ {}_ Gt = (s' \circ s)_{F''} \circ {}_ G(t' \circ t).$

According to our result above applied to the middle composition we may rewrite the left hand side as $s'_{F''} \circ s_{F''} \circ {}_ Gt' \circ {}_ Gt$ which is easily shown to be equal to the right hand side. $\square$

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