MATL, 16 bytes

t&=?1h}t2#X<wQw(

Try it online! Or verify all test cases

###How it works

t         % Take input implicitly. Duplicate
&=        % Matrix of all pairwise equality comparisons
?         % If all comparisons were true
  1h      %   Append 1 to the original copy ofthe array
}         % Else
  t       %   Duplicate array
  2#X<    %   Push minimum and index of its first occurrence
  wQw     %   Swap, increment, swap (adds 1 to the minimum)
  (       %   Assign the incremented minimum to that position
          % End if implicitly. Display implicitly

MATL, 16 bytes

t&=?1h}t2#X<wQw(

Try it online! Or verify all test cases

How it works

t         % Take input implicitly. Duplicate
&=        % Matrix of all pairwise equality comparisons
?         % If all comparisons were true
  1h      %   Append 1 to the original copy ofthe array
}         % Else
  t       %   Duplicate array
  2#X<    %   Push minimum and index of its first occurrence
  wQw     %   Swap, increment, swap (adds 1 to the minimum)
  (       %   Assign the incremented minimum to that position
          % End if implicitly. Display implicitly

MATL, 16 bytes

t&=?1h}t2#X<wQw(

Try it online! Or verify all test cases

###How it works

t         % Take input implicitly. Duplicate
&=        % Matrix of all pairwise equality comparisons
?         % If all comparisons were true
  1h      %   Append 1 to the original copy ofthe array
}         % Else
  t       %   Duplicate array
  2#X<    %   Push miximum and index of its first occurrence
  wQw     %   Swap, increment, swap (adds 1 to the minimum)
  (       %   Assign the incremented minimum to that position
          % End if implicitly. Display implicitly

MATL, 16 bytes

t&=?1h}t2#X<wQw(

Try it online! Or verify all test cases

###How it works

t         % Take input implicitly. Duplicate
&=        % Matrix of all pairwise equality comparisons
?         % If all comparisons were true
  1h      %   Append 1 to the original copy ofthe array
}         % Else
  t       %   Duplicate array
  2#X<    %   Push minimum and index of its first occurrence
  wQw     %   Swap, increment, swap (adds 1 to the minimum)
  (       %   Assign the incremented minimum to that position
          % End if implicitly. Display implicitly

MATL, 16 bytes

t&=?1h}t2#X<wQw(

Try it online! Or verify all test cases

MATL, 16 bytes

t&=?1h}t2#X<wQw(

Try it online! Or verify all test cases

###How it works

t         % Take input implicitly. Duplicate
&=        % Matrix of all pairwise equality comparisons
?         % If all comparisons were true
  1h      %   Append 1 to the original copy ofthe array
}         % Else
  t       %   Duplicate array
  2#X<    %   Push miximum and index of its first occurrence
  wQw     %   Swap, increment, swap (adds 1 to the minimum)
  (       %   Assign the incremented minimum to that position
          % End if implicitly. Display implicitly