In Jiri Rosicky's recent paper Metric monads, he claims in the proof of Proposition 5.1 that reflexive coequalizers in the category $\mathsf{PMet}$ of (generalized/extended) pseudo-metric spaces have the following description. Given a parallel pair $f, g : A \rightrightarrows B$ in $\mathsf{PMet}$ with a common section $t : B \to A$, one takes the coequalizer $q : B \to B/{\sim}$ of the underlying parallel pair in $\mathsf{Set}$ (where $\sim$ is the equivalence relation generated by all the pairs $(f(a), g(a))$ for $a \in A$), and then defines a pseudo-metric on the set $B/{\sim}$ by setting $$d([b_1], [b_2]) := \mathsf{inf}\{d(x, y) : x \sim b_1, y \sim b_2\}.$$
As I understand, this (attempted) definition of a pseudo-metric does not generally satisfy the triangle inequality (when $\sim$ is an arbitrary equivalence relation on the pseudo-metric space $B$), and so one has to resort to a more involved definition of the pseudo-metric on the quotient set (see e.g. https://en.wikipedia.org/wiki/Metric_space#Quotient_metric_spaces). Assuming Rosicky's description of reflexive coequalizers in $\mathsf{PMet}$ given above is correct, can someone explain why it works?