It is interesting to note that the general theory of operads with constants and that of operads without constants (here I refer to $O(0)$ as $\it{constants}$ showing my personal preference for terminology) admit the following distinct difference. Just for simplicity, let's consider operads enriched in sets and let's allow all (coloured) operads instead of just the monochromatic ones. Thus, let $\mathbf{Ope}$ be the category of all small coloured operads (symmetric or not, does not matter for this example) in $\mathbf{Set}$. Let $\mathbf{cfOpe}$ be the full subcategory of $\mathbf{Ope}$ consisting of the constant-free operads (that is those operads in which no $0$-ary arrows exist).

Now consider the obvious functors $j:\mathbf{Cat}\to\mathbf{Ope}$ and $l:\mathbf{Cat}\to\mathbf{cfOpe}$. It is rather simple to show that each of these functors has a right adjoint so we get $j':\mathbf{Ope}\to\mathbf{Cat}$ and $l': \mathbf{cfOpe}\to\mathbf{Cat}$. However, $l'$ has again a right adjoint while $l$ does not.

Not much changes if one considers operads enriched in some symmetric monoidal category $V$.

It is interesting to note that the general theory of operads with constants and that of operads without constants (here I refer to $O(0)$ as $\it{constants}$ showing my personal preference for terminology) admit the following distinct difference. Just for simplicity, let's consider operads enriched in sets and let's allow all (coloured) operads instead of just the monochromatic ones. Thus, let $\mathbf{Ope}$ be the category of all small coloured operads (symmetric or not, does not matter for this example) in $\mathbf{Set}$. Let $\mathbf{cfOpe}$ be the full subcategory of $\mathbf{Ope}$ consisting of the constant-free operads (that is those operads in which no $0$-ary arrows exist).

Now consider the obvious functors $j:\mathbf{Cat}\to\mathbf{Ope}$ and $l:\mathbf{Cat}\to\mathbf{cfOpe}$. It is rather simple to show that each of these functors has a right adjoint so we get $j':\mathbf{Ope}\to\mathbf{Cat}$ and $l': \mathbf{cfOpe}\to\mathbf{Cat}$. However, $l'$ has again a right adjoint while $j'$ does not.

Not much changes if one considers operads enriched in some symmetric monoidal category $V$.