Corollary

Under the hypotheses of Proposition 6, a subset $F$ of $X$ is closed in the topology $\mathfrak{T}$ if and only if $\overline{f}_i^{-1}(F)$ is closed in $Y_i$ for each $i \in I$.

This follows from the definition of the open sets in the topology $\mathfrak{T}$ by taking complements.

The general properties of final structures (Set Theory, Chapter IV, § 2, no. 5, criterion CST 19) imply the following transitivity property (the direct proof of which is also straightforward):