In Awodey's Category Theory the congruence category is defined as follows...
We have a congruence ~ on a category $C$. Then $C^\tilde{}$ is defined as:
$(C^\tilde{})_0=C_0$
$(C^\tilde{})_1=\{ \left<f,g\right>,f $~$ g \}$
$\tilde{1}_C=\left<1_C,1_C\right>$
$\left<f^\prime,g^\prime\right>\circ\left<f,g\right>=\left<f^\prime f,g^\prime g\right>$
Now what bothers me is that in general we may not have products in the category $C$, but the definition uses products of the form $A\times A$, where $A\in C_0$ (as far as I understood).
Any clarifications will be greatly appreciated :)
If I've understood correctly, you are concerned about the use of pairs $\langle f, g \rangle$, which you are thinking of as the product of those morphisms, when the category may not have products.
This is simply confusion caused by a clash of notation. The product of morphisms may not exist in the categorical sense, but Awodey is assuming that the category is locally small, so each collection of morphisms is a set, and one can always take the product of two sets. Here $\langle f, g \rangle$ just means a pair of morphisms, and $\{ \langle f, g \rangle | f \sim g \}$ means the set of all pairs of morhisms $f$ and $g$ such that $f$ is equivalent to $g$.