There are slide chain maps 
which are invertible up to homotopy. The inverses are given by the chain maps 
Proof.
The proof is given by an explicit calculation. As an example (the remaining cases are checked analogously) we show that \(\mathrm{slide}^{-1}_{\mathbf{1}_1,B_1}\circ\; \mathrm{slide}_{\mathbf{1}_1,B_1}\) is homotopic to the identity by computing their difference and exhibiting an explicit null-homotopy: 
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Original source: arXiv:2401.02956v2
Original source · 2401.02956v2