Showing $\Theta_{i}$ defined on $\prod _{i\in I}\mathcal{A}_{i}$ is a congruence relation.

Question

I am trying to use the previously asked question to help me finish my question. But I have a simple question about their "case 1" (when $a = b$). Why is it that if $a=b$, then $f(a)=f(b)$? Is this by definition or is it assumed f is well-defined? Thanks in advance.