I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf).
I try to understand why the amalgamated sum of two monoid morphisms $P \rightarrow Q_i$ possesses a presentation as a quotient of the direct sum:
$$Q_1\oplus_P Q_2 = Q_1\oplus Q_2/E$$
with $E$ being a congruence relation on $Q_1\oplus Q_2$. In Proposition 1.1.4 it is shown how the relation looks like, but in the proof it is already taken for granted that the quotient exists. There is speak of a natural map $Q_1\oplus Q_2 \rightarrow Q_1\oplus_P Q_2$, which I can't comprehend. Why does such a presentation exist?
As long as you have a congruence relation $\sim$ on a monoid $M$, the quotient monoid $M/{\sim}$ is perfectly defined. Look for semigroup congruence or quotient semigroup in the wikipedia entry Semigroup.