The usual definition on sets of a Cartesian product is $$ A \times B = \{ (a,b) | a \in A \land b \in B \} $$ Now we want to reason about products in a function like manner. To do this we introduce projections $$ \pi_1 : A \times B \rightarrow A $$ and $$ \pi_2 : A \times B \rightarrow B $$ If we consider the set of all tuples of the form $(X,f_1, f_2)$, we find that $(A \times B,\pi_1, \pi_2)$ is optimal in the following sense. For some set $C$ let $f : C \rightarrow A$ and $g : C \rightarrow B$. Now we can form the product function $$ \langle f,g \rangle : C \rightarrow A \times B $$ given as $$ \langle f,g \rangle (x) = (f(x),g(x)) $$ We can recover the functions of the product as thus $$ f = \pi_1 \circ \langle f , g \rangle $$ and $$ g = \pi_2 \circ \langle f , g \rangle $$ A quick proof of the first $$ (\pi_1 \circ \langle f, g \rangle)(x) = \pi_1(f(x),g(x)) = f(x) $$
We have other optimal choices. That is $(B \times A,\pi_2,\pi_2)$ is just as good. But these alternatives are isomorphic to each other.
A product of two objects is an object $A \times B$ together with the two projection arrows $$ \pi_1 : A \times B \rightarrow A $$ and $$ \pi_2 : A \times B \rightarrow B $$ such that for any object $C$ and pair of arrows $f : C \rightarrow A$ and $g : C \rightarrow B$, there is exactly one mediating arrow $$ \langle f,g \rangle : C \rightarrow A \times B $$ such that $$ \pi_1 \circ \langle f,g \rangle = f $$ and $$ \pi_2 \circ \langle f,g \rangle = g $$
If a category $\mathcal{C}$ has a product $A \times B$ for every pair of objects, $A$ and $B$, we say that $\mathcal{C}$ has all binary products.
If $A \times C$ and $B \times D$ are product objects, then for every pair of arrows, $f : A \rightarrow B$ and $g : C \rightarrow D$, the product map $$ f \times g : A \times C \rightarrow B \times D $$ is the arrow $$ \langle f \circ \pi_1 , g \circ \pi_2 \rangle $$
A coproduct of two objects, $A$ and $B$, is an object $A + B$ and two injection arrows $$ \iota_1 : A \rightarrow A + B $$ and $$ \iota_2 : B \rightarrow A + B $$ such that for any object $C$ and pair of arrows, $f : A \rightarrow C$ and $g : B \rightarrow C$, there is exactly one mediating arrow $$ [f,g] : A + B \rightarrow C $$ such that $$ [f,g] \circ \iota_1 = f $$ and $$ [f,g] \circ \iota_2 = g $$