  Now, the identity functor is related in a natural manner to the power set functor described above. Indeed, given a set \(X\) and its power set \(\wp(X)\), there is a function \(\mathbf{h}_X\) which takes an element \(x\) of \(X\) and sends it to the singleton set \(\{x\}\), a subset of \(X\), i.e., an element of \(\wp(X).\) This function in fact belongs to a family of functions indexed by the objects of the category of sets \(\{\mathbf{h}_Y : Y \rightarrow \wp(X)| Y \text{ in } \mathbf{Ob(Set)}\}.\) Moreover, it satisfies the following commutativity condition: given any function \(\boldsymbol{f}: X \rightarrow Y\), the identity functor yields the same function \(\boldsymbol{Id}(\boldsymbol{f}): \boldsymbol{Id}(X) \rightarrow \boldsymbol{Id}(Y).\) The commutativity condition thus becomes: \(\mathbf{h}_Y \circ \boldsymbol{Id}(\boldsymbol{f}) = \wp(\boldsymbol{f})  \circ \mathbf{h}_X.\) Thus the family of functions \(\boldsymbol{h}({\text{-}})\) relates the two functors in a natural manner. Such families of morphisms are called natural transformations between functors. Similarly, natural transformations between models of a theory yield the usual homomorphisms of structures in the traditional set theoretical framework.