Definition 4.28.1. Given a diagram as in the left hand side of:

\[ \xymatrix{ \mathcal{A} \rtwocell ^ F_{F'}{t} & \mathcal{B} \rtwocell ^ G_{G'}{s} & \mathcal{C} } \text{ gives } \xymatrix{ \mathcal{A} \rrtwocell ^{G \circ F} _{G' \circ F'}{\ \ s \star t} & & \mathcal{C} } \]

we define the *horizontal* composition $s \star t$ to be the transformation of functors ${}_{G'}t \circ s_ F = s_{F'}\circ {}_ Gt$.

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