# Kerodon

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Proposition 2.4.6.10. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $\sigma _0: \operatorname{\partial }\Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be a map of simplicial sets, which we identify with a diagram

$\xymatrix { & X_1 \ar [dr]^{ f_{21} } & \\ X_0 \ar [ur]^{ f_{10} } \ar [rr]_{ f_{20} } & & X_2 }$

as above. Then the construction of Example 2.4.3.10 induces a bijection

$\xymatrix { \{ \textnormal{Maps \sigma : \Delta ^2 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) with \sigma |_{ \operatorname{\partial }\Delta ^2} = \sigma _0} \} \ar [d]^{\sim } \\ \{ \textnormal{Homotopies from f_{21} \circ f_{10} to f_{20}} \} . }$

Proof of Proposition 2.4.6.10. Apply Corollary 2.4.6.13 in the case $n = 2$. $\square$