Define for posets with order $latex \sqsubseteq$: $latex \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigsqcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$; $latex \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigsqcap \{ x \in \mathfrak{B} \mid f x…
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