I was reading Mitchel book on categories and the following observation without proof is given:
A family of morphisms given by $\lbrace p_{i}:A \to A_{i} \rbrace$ is the product of $A_{i}$ if and only if $$ \phi_{B}:[B,A]_{\mathcal{C}} \to \prod_{i \in I}[B,A_{i}]_{\mathcal{C}}$$
where $\phi_{B}(\beta)=(p_{i},\beta)_{i \in I}$ for every $\beta \in [B,A]_{\mathcal{C}}$ is bijective for every object $\beta \in \mathcal{C}$, where $\prod_{i \in I}[B,A_{i}]_{\mathcal{C}}$ is the cartesian products on sets.
Can anyone help me understand this equivalence over the categorical definition of product in a category? How can I show by one side that $\phi_{B}(\beta)=(p_{i},\beta)_{i \in I}$ for every $\beta \in [B,A]_{\mathcal{C}}$ is bijective ? And how can this bijection give me the categorical product $\lbrace p_{i}:A \to A_{i} \rbrace$ of $A_{i}$ Can anyone help me giving a glimpse in order to look this statemente clearly?
Just write down everything carefully... There's no funny business going on.
The map $\phi_B$ is surjective: this means that for every element $(\beta_i) \in \prod_i [B,A_i]$, i.e. for every collection of morphisms $\beta_i : B \to A_i$, there exists some $\beta : B \to A$ such that $\phi_B(\beta) = (\beta_i)$. In other words, there exists a map $\beta : B \to A$ such that $p_i \circ \beta = \beta_i$. This is the first part of the universal property of a product.
And the map $\phi_B$ is also injective, meaning that if $\beta, \beta' : B \to A$ are such that $p_i \circ \beta = p_i \circ \beta'$ for all $i$, then $\beta = \beta'$.
If you combine the two, you get that for every collection of morphisms $(\beta_i : B \to A_i)_i$, there exists (part 1) a unique (part 2) morphism $\beta : B \to A$ such that $p_i \circ \beta = \beta_i$. This is exactly the definition of $(p_i : A \to A_i)_i$ being the product of the collection $(A_i)_i$.