I am going through Homotopy theory from the following book :
https://archive.org/details/borisovichbliznyakovizrailevichfomenkointroductiontotopologymir/mode/2up
At page 125, of this book , The homotopy group of a space gets introduced .
And it starts with an appeal to the reader to note that for every topological space , $\ Y $ , and continuous mapping $ f : X_1 \rightarrow X_2 $ of topological spaces there corresponds the natural mapping : $\pi^y(f) : \pi(X_2,Y) - \pi(X_1,Y)$
I could not understand what this $\pi $ mapping is as maybe because I am starting directly from homotopy in this book, but I tried searching for this mapping in whole book but could not find it. Maybe I missed it in the previous contents of the book.
I guess this is a generally used notion in mathematics. So if it is a general notion, what exactly is the definition of this pi mapping. Thanks in advance.
The map $\pi^Y(f)$ is a mapping which takes some equivalence class of maps $[\phi]: X_2 \rightarrow Y$ to the equivalence class maps $[\phi \circ f]: X_1 \rightarrow Y$. Of course it is left to show that this map is actually well defined i.e. that for $[\phi] = [\sigma]$ in $\pi(X_2, Y)$ we have $[\phi \circ f] = [\sigma \circ f]$ in $\pi(X_1, Y)$.
This is a special case of a (contravariant) functor between categories. Here between the (naive) homotopy category of topological spaces and sets. In fact, it is a special instance of a hom-functor. In general, let $\mathcal{A}$ be a category. Then a (contravariant) hom-functor $F_Y$ (based at an object $Y \in \mathcal{A}$) is a functor $\mathcal{A} \rightarrow \mathbf{Set}$ s.t. for any object $B \in \mathcal{A}$ we have $F_Y(B) = \text{Hom}(B,Y)$. On a morphism $f: A \rightarrow B$, the image under functor $F_Y(f)$ is a morphism $F_Y(B) = \text{Hom}(B,Y) \rightarrow \text{Hom}(A,Y) = F_Y(A)$. It takes an element $\phi \in \text{Hom}(B,Y)$, that is a map $\phi: B \rightarrow Y$ to a map $\phi \circ f: A \rightarrow Y$ i.e. an element in $\text{Hom}(A,Y)$.