A complete locally small category with no small coseparating set

A morphism $x\to y$ is by construction a set-indexed family of functions on the set $\mathrm{dom}(x)\cup \mathrm{dom}(y)$, so the morphisms $x\to y$ form a subset of the product of the function sets $Y_{\alpha }^{X_{\alpha }}$ over that index set. That product is a set, so every hom-collection is a set and $\mathcal{S}$ is locally small.

Let $D$ be a small diagram in $\mathcal{S}$ and let $T$ be the union of the supports of its set of objects, itself a set. Form the limit coordinatewise in $\mathbf{S}\mathbf{e}\mathbf{t}$: at $\alpha \in T$ take the limit $L_{\alpha }$ of the diagram of coordinates, and at $\alpha \mathrm{\notin }T$ every coordinate is $\{\mathrm{\varnothing }\}$, so the diagram there is constant at a one-element set and its limit is a one-element set. The code with domain $\{\alpha \in T:L_{\alpha }\text{ is not a singleton}\}$ and value $L_{\alpha }$ there is an object of $\mathcal{S}$, and its cone legs are the coordinatewise limit projections, the unique map being taken outside $T$. A cone over $D$ in $\mathcal{S}$ is exactly a coordinatewise cone, and the mediating map is coordinatewise unique, so this is a limit of $D$. The empty diagram has $T=\mathrm{\varnothing }$ and gives the code with empty domain. Every small diagram therefore has a limit, so [L2] makes $\mathcal{S}$ complete.

Let $\mathcal{H}$ be any small set of objects of $\mathcal{S}$. The union $T={\bigcup }_{H\in \mathcal{H}}\mathrm{dom}(H)$ is a set of ordinals. Were every ordinal a member of $T$ the ordinals would be a set, contradicting [L3], so some ordinal lies outside $T$; let $\beta$ be the least one, which is definable from $T$ and involves no selection.

Let $x$ be the code with empty domain, so $X_{\alpha }=\{\mathrm{\varnothing }\}$ for every $\alpha$, and let $y$ be the code with domain $\{\beta \}$ and $y(\beta )=\{0,1\}$. The two families $f,g:x\to y$ that send the single element of $X_{\beta }$ to $0$ and to $1$ respectively, and take the unique map at every other coordinate, are distinct morphisms. For every $H\in \mathcal{H}$ and every $h:y\to H$, the coordinate $H_{\beta }$ is $\{\mathrm{\varnothing }\}$ because $\beta \mathrm{\notin }\mathrm{dom}(H)$, so $h_{\beta }\circ f_{\beta }=h_{\beta }\circ g_{\beta }$; the two composites agree at every other coordinate as well, so $h\circ f=h\circ g$.

Therefore $\mathcal{H}$ does not detect the pair $f\mathrm{\ne }g$ and is not coseparating by [L1]. Since the argument applies to every small $\mathcal{H}$, $\mathcal{S}$ has no small coseparating set.