Mayr then describes how a function shift works as follows:

During this kind of functional shift, a structure always goes through a phase in which it can simultaneously perform two functions, such as Daphnia’s antennas, which are both a sensory organ and a floating rudder. This functional duality is possible because the genotype is a highly complex system that always produces certain aspects of the phenotype that are not directly promoted by selection but are simply “by-products” of the genotype favoured by selection.

Such by-products are then available for the acquisition of new functions. They are the ones that allow the front limbs (with a flying membrane) of a tetrapod to act as wings, or the lungs of a fish, as a swim bladder. In the phenotype of every organism there are numerous “neutral aspects” that are “admitted” by natural selection i.e. not eliminated but which have also not been specifically favoured by it.

These kinds of components of the phenotype are available for the transfer of new functions. Functional shifts are also known in macromolecules and behavioural patterns, for example, when plumage cleaning becomes part of advertising behaviour in certain ducks.
(Mayr 2002, p. 491)

Mayr continues on the intensification of the function of an existing organ:

As Severtsow has shown, it is often only necessary to intensify the function to enable a structure to take on what is apparently a new function. In this manner, for example, the front extremity of a walking mammal is transformed into the shovel of a mole, the wing of a bat or the fins of a whale.

The starting point for the development of eyes requires nothing more than the existence of light-sensitive cells. Natural selection already favours the acquisition of all necessary auxiliary mechanisms. That is the reason why photoreceptors or eyes have developed more than forty times independently of each other in the animal kingdom (…)

Nevertheless, the decisive factor in the acquisition of most evolving innovations is the change in behaviour.
(Mayr 2002, p. 491)

I quoted Ernst Mayr in such detail because I believe it is important to make clear how evolutionary innovations arise and how they do not. Evolutionary innovations do not arise from the fact that a functional organ is cultivated in some kind of fertile ground over many non-functional intermediate stages.

Evolutionary innovations only arise when an existing organ takes on a new task by shifting or enhancing its functionality. The conclusions for our question as to how the human mind arose are explained in hypothesis 5.