There is, indeed, something general under the carpet. I would have preferred a comment but I needed some more space.

1. Whenever you are able to to produce a functor $f:\Delta\to Something$ and "something" is a cocomplete category, then the functor you define ($X$ goes to the simplicial set $Map(\Delta[\bullet], X)$) has a left adjoint, obtained as the left Yoneda extension of your original $f$.

2. This mimicks (and in fact generalizes) the classical topological realization of a simplicial set, starting from how this should be defined on simplices, and then extending by cocontinuity, using that every presheaf is a colimit of representables

3. This has plenty of declinations in algebraic topology and category theory, so I'm not surprised it can be adapted to algebro-geometric settings! See in particular these discussions.

There is, indeed, something general under the carpet. I would have preferred a comment but I needed some more space.

1. Whenever you are able to to produce a functor $f:\Delta\to Something$ and "something" is a cocomplete category, then the functor you define ($X$ goes to the simplicial set $Map(\Delta[\bullet], X)$) has a left adjoint, obtained as the left Yoneda extension of your original $f$.

2. This mimicks (and in fact generalizes) the classical topological realization of a simplicial set, starting from how this should be defined on simplices, and then extending by cocontinuity, using that every presheaf is a colimit of representables

3. This has plenty of declinations in algebraic topology and category theory, so I'm not surprised it can be adapted to algebro-geometric settings! See in particular these discussions.

There is, indeed, something general under the carpet. I would have preferred a comment but I needed some more space.

1. Whenever you are ablto to produce a functor $f:\Delta\to Something$ and "something" is a cocomplete category, then the functor you define ($X$ goes to the simplicial set $Map(\Delta[\bullet], X)$) has a left adjoint, obtained as the left Yoneda extension of your original $f$.

2. This mimicks (and in fact generalizes) the classical topological realization of a simplicial set, starting from how this should be defined on simplices, and then extending by cocontinuity, using that every presheaf is a colimit of representables

3. This has plenty of declinations in algebraic topology and category theory, so I'm not surprised it can be adapted to algebro-geometric settings! See in particular these discussions.

There is, indeed, something general under the carpet. I would have preferred a comment but I needed some more space.

1. Whenever you are able to to produce a functor $f:\Delta\to Something$ and "something" is a cocomplete category, then the functor you define ($X$ goes to the simplicial set $Map(\Delta[\bullet], X)$) has a left adjoint, obtained as the left Yoneda extension of your original $f$.

2. This mimicks (and in fact generalizes) the classical topological realization of a simplicial set, starting from how this should be defined on simplices, and then extending by cocontinuity, using that every presheaf is a colimit of representables

3. This has plenty of declinations in algebraic topology and category theory, so I'm not surprised it can be adapted to algebro-geometric settings! See in particular these discussions.