1. This statement is famously not true for $\mathbf{Cat}$-enriched categories (strict 2-categories). The counterexample presented as 3.54 in Kelly's book (for the dual statement about limits) is based on the observation that the free 2-category generated by a single 2-cell $\alpha\colon f \Rightarrow f$ admits an underlying (1-)category which looks like $0 \xrightarrow{f} 1$, and that category admits $1 \times 1$ as a product, which should lift to a conical 2-limit but doesn't. This counterexample does not apply to $\mathbf{Pos}$-categories because the 2-cell structure is thin: an endo 2-cell must be the trivial identity 2-cell. I actually think I cannot build any counterexamples this way, because the functor $V = \mathbf{Pos}(\{*\}, -)\colon \mathbf{Pos} \to \mathbf{Set}$ is conservative (?) and hence any limit in the underlying category of a $\mathbf{Pos}$-category lifts to a conical $\mathbf{Pos}$-limit.

2. Riehl says in her book (I believe) that this is true for $\mathbf{Set}$ in the dual case precisely because a functor is representable if and only if its category of elements has an initial object (7.1.12). I haven't thought too hard about whether or not this is true for $\mathbf{Pos}$, but even if it were I don't quite follow her argument in the $\mathbf{Set}$ case; she seems to be saying that a weighted limit is given by an end expression, which in $\mathbf{Set}$ can be rewritten as a particular limit which looks like it could be the result of writing down a different limit (the one in the category of elements) as products and coequalisers, but I'm not sure how this generalises to $\mathcal{V}$-enriched or $\mathbf{Pos}$-enriched.

3. Every $\mathcal{V}$-enriched presheaf $\mathcal{C}^{\mathop{op}} \xrightarrow{F} \mathcal{V}$ is a weighted colimit of $\mathcal{V}$-representable functors. Section 3.9 of Kelly's book (I believe) presents an argument that if one were to try to write down an expression for $F$ using just conical $\mathcal{V}$-colimits, this map would only go one way in general (this is the comparison map of 3.56) - it just so happens that for the case of $\mathcal{V} = \mathbf{Set}$ this is an isomorphism. This argument feels almost a bit circular, so at least I cannot extract an intuitive explanation from it.

4. All $\mathcal{V}$-limits are built from conical $\mathcal{V}$-limits and cotensors. So the statement about ordinary categories can be thought of as $\mathbf{Set}$ having trivial cotensors. Indeed, in the $\mathbf{Cat}$-enriched case you only need a few different cotensors to get everything - I believe this is what PIE limits are about, the central idea of which is that all weighted $\mathbf{Cat}$-limits are given by products, inserters, and equifiers (with some caveats about strictness). I don't know what cotensors look like in $\mathbf{Pos}$ though, or how that would constitute a (dis)proof for my question.

1. This statement is famously not true for $\mathbf{Cat}$-enriched categories (strict 2-categories). The counterexample presented as 3.54 in Kelly's book (for the dual statement about limits) is based on the observation that the free 2-category generated by a single 2-cell $\alpha\colon f \Rightarrow f$ admits an underlying (1-)category which looks like $0 \xrightarrow{f} 1$, and that category admits $1 \times 1$ as a product, which should lift to a conical 2-limit but doesn't. This counterexample does not apply to $\mathbf{Pos}$-categories because the 2-cell structure is thin: an endo 2-cell must be the trivial identity 2-cell. I actually think I cannot build any counterexamples this way, because the functor $V = \mathbf{Pos}(\{*\}, -)\colon \mathbf{Pos} \to \mathbf{Set}$ is conservative (?) and hence any limit in the underlying category of a $\mathbf{Pos}$-category lifts to a conical $\mathbf{Pos}$-limit.

2. Riehl says in her book (I believe) that this is true for $\mathbf{Set}$ in the dual case precisely because a functor is representable if and only if its category of elements has an initial object (7.1.12). I haven't thought too hard about whether or not this is true for $\mathbf{Pos}$, but even if it were I don't quite follow her argument in the $\mathbf{Set}$ case; she seems to be saying that a weighted limit is given by an end expression, which in $\mathbf{Set}$ can be rewritten as a particular limit which looks like it could be the result of writing down a different limit (the one in the category of elements) as products and equalisers, but I'm not sure how this generalises to $\mathcal{V}$-enriched or $\mathbf{Pos}$-enriched.

3. Every $\mathcal{V}$-enriched presheaf $\mathcal{C}^{\mathop{op}} \xrightarrow{F} \mathcal{V}$ is a weighted colimit of $\mathcal{V}$-representable functors. Section 3.9 of Kelly's book (I believe) presents an argument that if one were to try to write down an expression for $F$ using just conical $\mathcal{V}$-colimits, this map would only go one way in general (this is the comparison map of 3.56) - it just so happens that for the case of $\mathcal{V} = \mathbf{Set}$ this is an isomorphism. This argument feels almost a bit circular, so at least I cannot extract an intuitive explanation from it.

4. All $\mathcal{V}$-limits are built from conical $\mathcal{V}$-limits and cotensors. So the statement about ordinary categories can be thought of as $\mathbf{Set}$ having trivial cotensors. Indeed, in the $\mathbf{Cat}$-enriched case you only need a few different cotensors to get everything - I believe this is what PIE limits are about, the central idea of which is that all weighted $\mathbf{Cat}$-limits are given by products, inserters, and equifiers (with some caveats about strictness). I don't know what cotensors look like in $\mathbf{Pos}$ though, or how that would constitute a (dis)proof for my question.

1. This statement is famously not true for $\mathbf{Cat}$-enriched categories (strict 2-categories). The counterexample presented as 3.54 in Kelly's book (for the dual statement about limits) is based on the observation that the free 2-category generated by a single 2-cell $\alpha\colon f \Rightarrow f$ admits an underlying (1-)category which looks like $0 \xrightarrow{f} 1$, and that category admits $1 \times 1$ as a product, which should lift to a conical 2-limit but doesn't. This counterexample does not apply to $\mathbf{Pos}$-categories because the 2-cell structure is thin: an endo 2-cell must be the trivial identity 2-cell. I actually think I cannot build any counterexamples this way, because the functor $V = \mathbf{Pos}(\{*\}, -)\colon \mathbf{Pos} \to \mathbf{Set}$ is conservative (?) and hence any limit in the underlying category of a $\mathbf{Pos}$-category lifts to a conical $\mathbf{Pos}$-limit.

2. Riehl says in her book (I believe) that this is true for $\mathbf{Set}$ in the dual case precisely because a functor is representable if and only if its category of elements has an initial object (7.1.12). I haven't thought too hard about whether or not this is true for $\mathbf{Pos}$, but even if it were I don't quite follow her argument in the $\mathbf{Set}$ case; she seems to be saying that a weighted limit is given by an end expression, which in $\mathbf{Set}$ can be rewritten as a particular limit which looks like it could be the result of writing down a different limit (the one in the category of elements) as products and coequalisers, but I'm not sure how this generalises to $\mathcal{V}$-enriched or $\mathbf{Pos}$-enriched.

3. Every $\mathcal{V}$-enriched presheaf $\mathcal{C}^{\mathop{op}} \xrightarrow{F} \mathcal{V}$ is a weighted colimit of $\mathcal{V}$-representable functors. Section 3.9 of Kelly's book (I believe) presents an argument that if one were to try to write down an expression for $F$ using just conical $\mathcal{V}$-colimits, this map would only go one way in general (this is the comparison map of 3.56) - it just so happens that for the case of $\mathcal{V} = \mathbf{Set}$ this is an isomorphism. This argument feels almost a bit circular, so at least I cannot extract an intuitive explanation from it.

4. All $\mathcal{V}$-limits are built from conical $\mathcal{V}$-limits and cotensors. So the statement about ordinary categories can be thought of as $\mathbf{Set}$ having trivial cotensors. Indeed, in the $\mathbf{Cat}$-enriched case you only need a few different cotensors to get everything - I believe this is what PIE limits are about, the central idea of which is that all weighted $\mathbf{Cat}$-limits are given by products, inserters, and equifiers (with some caveats about strictness). I don't know what cotensors look like in $\mathbf{Pos}$ though, or how that would constitute a (dis)proof for

Riehl, Emily, Categorical homotopy theory, New Mathematical Monographs 24. Cambridge: Cambridge University Press (ISBN 978-1-107-04845-4/hbk; 978-1-107-26145-7/ebook). xviii, 352 p. (2014). ZBL1317.18001. my question.

1. This statement is famously not true for $\mathbf{Cat}$-enriched categories (strict 2-categories). The counterexample presented as 3.54 in Kelly's book (for the dual statement about limits) is based on the observation that the free 2-category generated by a single 2-cell $\alpha\colon f \Rightarrow f$ admits an underlying (1-)category which looks like $0 \xrightarrow{f} 1$, and that category admits $1 \times 1$ as a product, which should lift to a conical 2-limit but doesn't. This counterexample does not apply to $\mathbf{Pos}$-categories because the 2-cell structure is thin: an endo 2-cell must be the trivial identity 2-cell. I actually think I cannot build any counterexamples this way, because the functor $V = \mathbf{Pos}(\{*\}, -)\colon \mathbf{Pos} \to \mathbf{Set}$ is conservative (?) and hence any limit in the underlying category of a $\mathbf{Pos}$-category lifts to a conical $\mathbf{Pos}$-limit.

2. Riehl says in her book (I believe) that this is true for $\mathbf{Set}$ in the dual case precisely because a functor is representable if and only if its category of elements has an initial object (7.1.12). I haven't thought too hard about whether or not this is true for $\mathbf{Pos}$, but even if it were I don't quite follow her argument in the $\mathbf{Set}$ case; she seems to be saying that a weighted limit is given by an end expression, which in $\mathbf{Set}$ can be rewritten as a particular limit which looks like it could be the result of writing down a different limit (the one in the category of elements) as products and coequalisers, but I'm not sure how this generalises to $\mathcal{V}$-enriched or $\mathbf{Pos}$-enriched.

3. Every $\mathcal{V}$-enriched presheaf $\mathcal{C}^{\mathop{op}} \xrightarrow{F} \mathcal{V}$ is a weighted colimit of $\mathcal{V}$-representable functors. Section 3.9 of Kelly's book (I believe) presents an argument that if one were to try to write down an expression for $F$ using just conical $\mathcal{V}$-colimits, this map would only go one way in general (this is the comparison map of 3.56) - it just so happens that for the case of $\mathcal{V} = \mathbf{Set}$ this is an isomorphism. This argument feels almost a bit circular, so at least I cannot extract an intuitive explanation from it.

4. All $\mathcal{V}$-limits are built from conical $\mathcal{V}$-limits and cotensors. So the statement about ordinary categories can be thought of as $\mathbf{Set}$ having trivial cotensors. Indeed, in the $\mathbf{Cat}$-enriched case you only need a few different cotensors to get everything - I believe this is what PIE limits are about, the central idea of which is that all weighted $\mathbf{Cat}$-limits are given by products, inserters, and equifiers (with some caveats about strictness). I don't know what cotensors look like in $\mathbf{Pos}$ though, or how that would constitute a (dis)proof for my question.

Riehl, Emily, Categorical homotopy theory, New Mathematical Monographs 24. Cambridge: Cambridge University Press (ISBN 978-1-107-04845-4/hbk; 978-1-107-26145-7/ebook). xviii, 352 p. (2014). ZBL1317.18001.