Composition functor

Construction 1.4.7 (Composition functor). Consider the zig-zag \[ \Ar (C) \times _{s,C,t} \Ar (C) \xleftarrow [\sim ]{(d_0^*,d_2^*)} \Fun ([2],C) \xrightarrow {d^*_1} \Ar (C). \] The first map is an equivalence, and hence it admits an inverse, providing a composite functor \[ - \circ - \colon \Ar (C) \times _{s,C,t} \Ar (C) \xrightarrow {\sim } \Fun ([2],C) \xrightarrow {d^*_1} \Ar (C). \] We refer to this as the composition functor.