We recall the notion of a $k$-space (or compactly generated space) to fix our notations. For every topological space $X$, we can define a category $\mathfrak{M}_X$. The class of objects of $\mathfrak{M}_X$ is the class of continuous mappings $u: K \to X$ from a compact Hausdorff space $K$ to $X$. Let $v : C \to X$ be another object in $\mathfrak{M}_X$, the set of morphisms from $u$ to $v$ is the continuous mappings $f: K \to C$ such that $u = v\circ f$. Let $\tau$ be the collection of open sets in $X$. Define $k(\tau)$ to be the final topology on $X$ with respect to the class $\mathrm{Obj}(\mathfrak{M}_X)$. In other words, for any subset $U$ of $X$, $U \in k(\tau)$ if and only if $u^{-1}(U)$ is open in $K$ for any object $u : K \to X$ in $\mathfrak{M}_X$. Let $k(X)$ be the space $X$ equipped with the topology $k(\tau)$ which is finer than $\tau$ by definition. Then we call $X$ a $k$-space if $k(X) = X$, or more precisely, $\tau = k(\tau)$.

If we adopt the approach of the Grothendieck's axiom on universe to approach the above categorical setting. We can fix a universe $\mathcal{U}$ with $X \in \mathcal{U}$. The category $\mathfrak{M}_X$ is redefined as the category of $\mathcal{U}$-small continuous mappings $u : K \to X$ from a compact Hausdorff $K$ which is also $\mathcal{U}$-small. To emphasize this dependence on our universe $\mathcal{U}$, we denote this category by $\mathfrak{M}_{X,\mathcal{U}}$. Similarly, the final topology on $X$ with respect to $\mathrm{Obj}(\mathfrak{M}_{X, \mathcal{U}})$ is denote by $k_{\mathcal{U}}(\tau)$.

My question is, if we have another universe $\mathcal{V}$ with $X \in \mathcal{V}$, does $k_{\mathcal{U}}(\tau) = k_{\mathcal{V}}(\tau)$?

A sufficient condition would be there is a small (small is for any universe containing $X$) final subcategory $\mathfrak{S}_X$ of $\mathfrak{M}_X$, as is the case when $X$ is assumed to be weakly Hausdorff (we can take $\mathfrak{S}_X$ to be the full subcategory of $\mathfrak{M}_X$ generated by all embeddings $K \hookrightarrow X$ where $K$ is a closed compact Hausdorff subspace of $X$). But for general topological space $X$, can we find a small final subcategory $\mathfrak{S}_X$ of $\mathfrak{M}_X$?