SCHWEITZER 2018 Problem 7

Describe all functions $f: \{ 0,1\}^n \to \{ 0,1\}$ which satisfy the equation \begin{align*} & f(f(a_{11},a_{12},\dotsc ,a_{1n}),f(a_{21},a_{22},\dotsc ,a_{2n}),\dotsc ,f(a_{n1},a_{n2},\dotsc ,a_{nn})) & = f(f(a_{11},a_{21},\dotsc ,a_{n1}),f(a_{12},a_{22},\dotsc ,a_{n2}),\dotsc ,f(a_{1n},a_{2n},\dotsc ,a_{nn}))\end{align*} for arbitrary $a_{ij}\in \{ 0,1\}$ where $i,j\in \{1,2,\dotsc ,n\}.$