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First one removes the condition that $S_\emptyset$ and $T_\emptyset$ are initial, which expands the category above a bit by replacing "suplattice" and "$\mathbb{N}$-set" by the categories obtained by freely adding initial objects to the category of suplattices and $\mathbb{N}$-sets.

Then we add the condition that $S_\emptyset \coprod T_1 = S_{1}$ and $F_\emptyset \coprod T_1 = F_{1}$ through the obvious maps. It is easy to check that these are indeed colimits in the category of representables, so this is still a realized sketch.

Now, the triples as above also satisfy that as soon as the suplattice $S$ is not this freely added initial object, then the map $S \to T$ is a bijection and similarly, as soon as the $F$ is not the freely added initial object, then the map $F \to T$ is a bijection.

This is exactly the category of triples described above.

Assume that I have a "realized" sketch $T$, which I see as a full subcategory of its category of models. I'm for simplicity assuming that T is stable under finite coproducts, and all cocone in T that are colimits in the category of models are marked cocone of the sketch.

Then If I consider the sketch T' with the same category as T where I've remove from the list of marked cocone, all the empty ones (that specify the initial objects). I claim that the category of T' model identify with the category of T-model together with a new initial object (the empty model).

Indeed, as all special cocone of T' are non-empty, one easily see that the empty model is a model of T'.

Now given any non-empty model X of T', i.e. such that X(a) is inhabited for some object $a$. In T', we have $a \coprod e \simeq a$ (where e is the initial object) and it corresponds to a marked cocone. So it follows that the projection map $X(a) \times X(e) \to X(a)$ is a bijection. But as $X(a)$ is inhabited, this implies that $X(e)=\{*\}$, and hence that $X$ is actually a $T$ model.

Given that all $T$-model are non-empty (as $X(e)=\{*\}$) we have that a $T'$ model is either the empty model or a $T$-model.

First one removes the condition that $S_\emptyset$ and $T_\emptyset$ are initial, which expands the category above a bit by replacing "suplattice" and "$\mathbb{N}$-set" by the categories obtained by freely adding initial objects to the category of suplattices and $\mathbb{N}$-sets (see the edit below for more details).

Then we add new marked cocone, implementing the condition that $S_\emptyset \coprod T_1 = S_{1}$ and $F_\emptyset \coprod T_1 = F_{1}$ through the obvious maps. It is easy to check that these are indeed colimits in the category of representables (i.e. in the sketch), so this is still a realized sketch.

Now, the triples as above are model of this new sketch if as soon as the suplattice $S$ is not the freely added initial object, then the map $S \to T$ is a bijection and similarly, as soon as the $F$ is not the freely added initial object, then the map $F \to T$ is a bijection.

This gives us exactly the category of triples $(X,A,B)$ described above ($A$ and $B$ just remembering if the "suplatice" or the "$\mathbb{N}$-Set" are the freely added initial object or not).

Assume that I have a "realized" sketch $T$, which I see as a full subcategory of its category of models. For simplicity, I'm assuming that T is stable under finite coproducts, and that all cocone in T that are colimits in the category of models are marked cocone of the sketch.

I now consider the sketch T' with the same underlying category as T, but where I have removed from the list of marked cocone all the empty ones (i.e. the cocone that would specify an initial object). I claim that the category of T' model identify with the category of T-model together with a new, freely added initial object given by the empty model.

Indeed, as all special cocone of T' are non-empty, one easily see that the empty model (sending each ovject of $T$ to the empty set) is a model of T'.

Now given any non-empty model X of T', i.e. such that X(a) is inhabited for some object $a$. In T', if I write $e$ for the initial object of $T'$, we have $a \coprod e \simeq a$ and it corresponds to a marked cocone $a,e \to a$. This mean that the unique map $e \to a$, induce a map $X(a) \to X(e)$ such that the map $X(a) \to X(a) \times X(e)$ is a bijection, in particular, the projection map $X(a) \times X(e) \to X(a)$ being a right inverse of it is also a bijection. But as $X(a)$ is inhabited, this implies that $X(e)=\{*\}$, and hence that $X$ is not justs a $T'$ model but actually a $T$ model as it is also compatible to the empty cocone of $T$.

Given that all $T$-model are non-empty (as $X(e)=\{*\}$) we have that a $T'$ model is either the empty model or a $T$-model. Hence the result.

Edit: let me details a side construction that I'm using which I think will clarify a bit what I'm doing here.

Assume that I have a "realized" sketch $T$, which I see as a full subcategory of its category of models. I'm for simplicity assuming that T is stable under finite coproducts, and all cocone in T that are colimits in the category of models are marked cocone of the sketch.

Then If I consider the sketch T' with the same category as T where I've remove from the list of marked cocone, all the empty ones (that specify the initial objects). I claim that the category of T' model identify with the category of T-model together with a new initial object (the empty model).

Indeed, as all special cocone of T' are non-empty, one easily see that the empty model is a model of T'.

Now given any non-empty model X of T', i.e. such that X(a) is inhabited for some object $a$. In T', we have $a \coprod e \simeq a$ (where e is the initial object) and it corresponds to a marked cocone. So it follows that the projection map $X(a) \times X(e) \to X(a)$ is a bijection. But as $X(a)$ is inhabited, this implies that $X(e)=\{*\}$, and hence that $X$ is actually a $T$ model.

Given that all $T$-model are non-empty (as $X(e)=\{*\}$) we have that a $T'$ model is either the empty model or a $T$-model.

So, in the resulting category of models, if you try to construct the pushout of $(\{0,1\},1,\emptyset)$ with $(\emptyset,\emptyset,1)$ you are constructing $(V,1,1)$ where $V$ is the initial ZF-algebra, which is always large (it is in bijection with the class of all sets as mentioned before).

So, in the resulting category of models, if you try to construct the coproduct of $(\{0,1\},1,\emptyset)$ with $(\emptyset,\emptyset,1)$ you are constructing $(V,1,1)$ where $V$ is the initial ZF-algebra, which is always large (it is in bijection with the class of all sets as mentioned before).