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Consider the terminal object $1\in\mathbf{Pos}$.

Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the copower (= tensor) of $1$ by $X$.

On the other hand, the closure of $1$ in $\mathbf{Pos}$ undel all conical colimits is the full subcategory spanned by the dicrete posets (which is equivalent to $\mathbf{Set}$); this is because every discrete poset is a coproduct of copies of $1$ and any colimit of discrete posets is still discrete.

Thus, it is not possible to express any $\mathbf{Pos}$-weighted colimit as a conical one (otherwise the two $\mathbf{Pos}$-enriched categories above would be the same).

This counterexample works more generally whenever the unit $I$ of the base of enrichment $\mathcal V_0$ is not colimit-dense (that is, the closure of $I$ under ordinary colimits in $\mathcal V_0$ is strictly contained in $\mathcal V_0$).

Consider the terminal object $1\in\mathbf{Pos}$.

Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the copower (= tensor) of $1$ by $X$.

On the other hand, the closure of $1$ in $\mathbf{Pos}$ undel all conical colimits is the full subcategory spanned by the dicrete posets (which is equivalent to $\mathbf{Set}$); this is because every discrete poset is a coproduct of copies of $1$ and any conical colimit of discrete posets is still discrete.

Thus, it is not possible to express any $\mathbf{Pos}$-weighted colimit as a conical one (otherwise the two $\mathbf{Pos}$-enriched categories above would be the same).

This counterexample works more generally whenever the unit $I$ of the base of enrichment $\mathcal V_0$ is not colimit-dense (that is, the closure of $I$ under ordinary colimits in $\mathcal V_0$ is strictly contained in $\mathcal V_0$).

Consider the terminal object $1\in\mathbf{Pos}$.

Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the copower (= tensor) of $1$ by $X$.

On the other hand, the closure of $1$ in $\mathbf{Pos}$ undel all conical colimits is the full subcategory spanned by the dicrete posets (which is equivalent to $\mathbf{Set}$); this is because any colimit of discrete posets is still discrete.

Thus, it is not possible to express any $\mathbf{Pos}$-weighted colimit as a conical one (otherwise the two $\mathbf{Pos}$-enriched categories above would be the same).

This counterexample works more generally whenever the unit $I$ of the base of enrichment $\mathcal V_0$ is not colimit-dense (that is, the closure of $I$ under ordinary colimits in $\mathcal V_0$ is strictly contained in $\mathcal V_0$).

Consider the terminal object $1\in\mathbf{Pos}$.

Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the copower (= tensor) of $1$ by $X$.

On the other hand, the closure of $1$ in $\mathbf{Pos}$ undel all conical colimits is the full subcategory spanned by the dicrete posets (which is equivalent to $\mathbf{Set}$); this is because every discrete poset is a coproduct of copies of $1$ and any colimit of discrete posets is still discrete.

Thus, it is not possible to express any $\mathbf{Pos}$-weighted colimit as a conical one (otherwise the two $\mathbf{Pos}$-enriched categories above would be the same).

This counterexample works more generally whenever the unit $I$ of the base of enrichment $\mathcal V_0$ is not colimit-dense (that is, the closure of $I$ under ordinary colimits in $\mathcal V_0$ is strictly contained in $\mathcal V_0$).

Consider the terminal object $1\in\mathbf{Pos}$.

Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the copower of $1$ by $X$.

On the other hand, the closure of $1$ in $\mathbf{Pos}$ undel all conical colimits is the full subcategory spanned by the dicrete posets (which is equivalent to $\mathbf{Set}$); this is because any colimit of discrete posets is still discrete.

Thus, it is not possible to express any $\mathbf{Pos}$-weighted colimit as a conical one (otherwise the two $\mathbf{Pos}$-enriched categories above would be the same).

Consider the terminal object $1\in\mathbf{Pos}$.

Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the copower (= tensor) of $1$ by $X$.

On the other hand, the closure of $1$ in $\mathbf{Pos}$ undel all conical colimits is the full subcategory spanned by the dicrete posets (which is equivalent to $\mathbf{Set}$); this is because any colimit of discrete posets is still discrete.

Thus, it is not possible to express any $\mathbf{Pos}$-weighted colimit as a conical one (otherwise the two $\mathbf{Pos}$-enriched categories above would be the same).

This counterexample works more generally whenever the unit $I$ of the base of enrichment $\mathcal V_0$ is not colimit-dense (that is, the closure of $I$ under ordinary colimits in $\mathcal V_0$ is strictly contained in $\mathcal V_0$).