Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of pairwise disjoint metric spaces. Here we use the convention that a metric space can assume infinite distance. Let Met be the category with metric spaces as objects and non-expansive maps as morphisms. Let $F:Top\to Met$ be the forgetful functor, where Top is the category of topological spaces as objects with continuous maps as morphisms.
Then, unless I'm mistaken, the coproduct $\coprod_{i \in \mathbb{N}}^{Met} X_i$ in Met is the set-theoretic coproduct with metric given by $$ d(f,g):=\begin{cases} d_i(f,g): & f,g \in X_i\\ \infty :& else \end{cases} , $$ where $d_i$ is the metric on $X_i$. Let $\coprod_{i\in \mathbb{N}}^{Top} F(X_i)$ denote the coproduct in Top. Is is the case that $$ \coprod_{i \in \mathbb{N}}^{Top} F(X_i) \cong F\left( \coprod_{i \in \mathbb{N}}^{Met} X_i \right) . $$ Or have I made a mistake somewhere?
This is correct. You metric induces the topology of the disjoint union of topological spaces.