\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] {$\sim$};