# Kerodon

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Corollary 4.5.4.14. Suppose we are given a commutative diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ A_0 \ar [d] & A \ar [l]_{f_0} \ar [r]^-{f_1} \ar [d] & A_1 \ar [d] \\ B_0 & B \ar [l]_{g_0} \ar [r]^-{g_1} & B_1, }$

where $f_0$ and $g_0$ are monomorphisms and the vertical maps are categorical equivalences. Then the induced map

$A_0 \coprod _{A} A_1 \rightarrow B_0 \coprod _{B} B_1$

is a categorical equivalence.