The Stacks project

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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