Sandbox

[ilmath]\begin{xy} \xymatrix{ I \ar[r]^f \ar[d] \ar@{.>}[dr] & J \ar[r]^g & K \\ A \ar[r]^-f & B\otimes B\otimes B \ar[r]^g & X \\ A \ar[r]^f & B\otimes B\otimes B \ar[r]^g & X \\ C \ar@(dr,dl)@2{_{(}->}[r] _a = "a" & D \ar@{_{(}-^{)} }[r]_b = "b" & Y %Extra arrows \ar@2{<-} "a";"b" } \end{xy}[/ilmath]

[ilmath]\begin{xy} %vertices (0,0)*+{E^{[0]}=B}="v1"; (100,0)*+{A=B}="v2"; {\ar@{->>}^{p_{n+1} } "v1"; "v2"}; \end{xy}[/ilmath]

[ilmath]\begin{xy}\xymatrix{ A \ar@2{->}[r] \ar@{->}[d] & B\otimes B\otimes B \ar@{^{(}~>}[d] \ar@/_/@{<.>}[r] \ar@/^/@{(-)}[r] \ar@2{`-)}[r] & L_\text{ast}\\ B \ar@/^/@{_{<}-^{>} }[r] \ar@{_{(}->}[r] & D_5 \ar@{_{(}->}[r] & C }\end{xy}[/ilmath]