A congruence category is defined as follows according to Awodey's book Category Theory:
We have a congruence $\sim$ on a category $C$. Then $C^\sim$ is defined as: \begin{align*} (C^\sim)_0 &= C_0 \\ (C^\sim)_1 &= \{ \langle f, g \rangle, f \sim g \} \\ \tilde{1}_C &= \langle 1_C, 1_C \rangle \\ \langle f', g' \rangle \circ \langle f, g \rangle &= \langle f' f, g' g \rangle \end{align*}
From this definition I cannot understand what the objects of $C^\sim$ should be. One possibility is that $C^\sim$ has the same objects as $C$, except for the terminal object $(C^\sim)_1$ which is already well defined as a set of arrow "pairs" (clarified here: https://math.stackexchange.com/a/1578265/587087).
Another confusion is the fact that Awodey is defining an object of $C^\sim$, $(C^\sim)_1$ as a set. Should I read this definition as the set of incoming arrows of the terminal object? In that case since congruence relation is an equivalence relation and equivalence relations are reflexive, we would have all the incoming arrows of $C_1$ also in $(C^\sim)_1$. With this approact, I would define objects and morphisms as follows:
A congruence category of the category $C$, $C^\sim$ has objects of $C$ as objects, and morphisms are defined as follows:
$$ \forall f\in \operatorname{Hom}(X, Y)_{C}: \langle f, f \rangle \in \operatorname{Hom}(X, Y)_{C^\sim} $$
Until now, we defined objects of newly constructed categories in terms of the base categories; we never defined them in such manner. So the question is that is my understanding of this definition of the congruence category accurate? Am I missing something here?
