my blog lives here now
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\node[] (i0j1) at (-1, 1) {A};
\node[] (i1j1) at (1, 1) {B};
\node[] (i0j0) at (-1, -1) {B};
\node[] (i1j0) at (1, -1) {B};
\draw[<-] (i0j0) -- (i0j1) node [midway] {$\mathrm{equiv}$} node [midway, above, rotate=-90] {$\sim$};
\draw[->] (i0j0) -- (i1j0) node [midway, below] {$B$};
\draw[->,dashed] (i0j1) -- (i1j1) node [midway] {};
\draw[<-] (i1j0) -- (i1j1) node [midway, right] {$\mathrm{id}_B$} node [midway, above, rotate=90] {$\sim$};