Does the 'Universal Property of Initial Topology' guarantee the existence of a continuous map $g$ such that $f_\alpha \circ g$ is continuous?

Question

This is a question to the first answer given on this thread: Product of path connected spaces is path connected. Unfortunately, I have too little experience to comment this directly below the answer.

Why does the 'universal property of the initial topology' (outlined below) yield the existence of a unique mapping $\gamma$ such that $\pi_i \circ \gamma = \gamma_i$? As in, why is it not necessary to define a mapping $\gamma$ and to check whether $\pi_i \circ \gamma$ holds for all $i$?

Proposition (Universal Property of the initial topology). Let $X$ be a set, let $(Y_\alpha)_{\alpha \in A}$ be a family of topological spaces, and let $f_\alpha \colon X \to Y_\alpha$ be functions. Let $\mathcal T_X$ be the topology on $X$. Then $\mathcal T_X$ is the initial topology on $X$ with respect to the maps $f_\alpha$ if and only if $(X, \mathcal T_X)$ has the following property: If $(Z, \mathcal T_Z)$ is a topological space and $g \colon Z \to X$ is a map, then $g$ is continuous if and only if $f_\alpha \circ g \colon Z \to Y_\alpha$ is continuous for all $\alpha$.

In other words, (since the question in the link discusses a situation with a product topology on $\prod_\alpha Y_\alpha$,) why does the above proposition yield the existence of a unique map $\gamma$ as given in the answer? I managed to define $\gamma$ as a map $X \to \prod_\alpha Y_\alpha$ given by $t \mapsto (\gamma_i(t))_{i \in I}$, such that indeed $\pi_i \circ \gamma = \gamma_i (=$ continuous), hence, by the proposition $\gamma$ must be continuous. But what ensures existence of $\gamma$ in the first place?

Answer 1

The universal property serves to give a criterion when a map $f: X \to Y$ into a space with the initial topology w.r.t. a set $\{f_\alpha: Y \to X_\alpha: \alpha \in A\}$ is continuous: we exactly need all compositions with $f_\alpha$ to be continuous. This is clearly a necessary condition, but when we use the initial topology it is sufficient too (the fact that this property itself characterises the initial topology among all topologies on $Y$ is irrelevant here).

This is very convenient for product spaces, (where $f_\alpha = \pi_\alpha$ ,the projection) as it's easy to make maps into a product that respect the projections in this way: e.g. when you define (this is just set theory) $\gamma(t)$ as having the $\alpha$'th coordinate equal to $\gamma_\alpha(t)$. So then by construction we have $\pi_\alpha \circ \gamma =\gamma_\alpha$ and as all $\gamma_\alpha$ are continuous, so is the newly defined $\gamma$, thanks to this criterion and the fact we defined $\gamma$ coordinate-wise..

So the criterion does not garantuee such a map, but set theory tells us that if we have a family of functions $g_\alpha: X \to Y_\alpha$ we can always define $\nabla_\alpha g_\alpha: X \to \prod_\alpha Y_\alpha$ by $\pi_\alpha \circ (\nabla_\alpha g_\alpha) = g_\alpha$, as a function on a product is uniquely determined by its projection values , and the criterion then gives that if the $Y_\alpha$ were spaces and we used the product topology this $\nabla_\alpha g_\alpha$ is then automatically continuous.