Kdjykuj

Let $mathcal{O}$ be an operad in the monoidal category $M$. Then $mathcal{O}(1)$ together with the morphisms
$$mathcal{O}(1)otimes mathcal{O}(1)to mathcal{O}(1)$$
and the unit $eta:1to mathcal{O}(1)$ is a monoid object. Moreover, a morphism $varphi:mathcal{O}to mathcal{O}'$ of operads induces a morphism $mathcal{O}(1)to mathcal{O}(1)$ of monoid objects. Therefore, we get a forgetful functor
$$mathrm{Operads}(M)to mathrm{Monoids}(M).$$
If we work with coloured operads, we get a functor from coloured $M$-operads to $M$-enriched categories. Conversely, if $T$ is a monoid object, we can build an operad by
$$mathcal{O}_T(r) := T^{otimes r}.$$
and the structure maps
$$T^{otimes r}otimes bigotimes_{i=1}^r T^{otimes k_i}to T^{otimes (k_1+dotsb+k_r)}$$
as follows: Let $Delta:Tto T^{otimes k}$ be the diagonal (existence is clear if $otimes$ is the categorical product). Then
$$Totimes T^{otimes k}stackrel{Deltaotimes mathrm{id}}{to}T^{otimes k}otimes T^{otimes k} cong (T^{otimes 2})^{otimes k} to T^{otimes k}.$$
In $mathbf{Set}$, this just means $t(t_1,dotsc,t_k)=(tt_1,dotsc,tt_k)$.
It should be clear that this construction gives us an operad. Now two problems/questions:

1. Does the morphism $Delta$ always exist in the general setting? It is obviously not the same as
   $$Tcong Totimes 1^{otimes (k-1)}stackrel{mathrm{id}otimes eta^{otimes (k-1)}}{to} T^{otimes k}.$$
   


2. Obviously $mathcal{O}_T(1)cong T$, but it seems to be not true that $mathcal{O}_T$ is the free operad over the monoid object $T$. Is there another construction for the “free” operad over $T$?

Let me mention that this is related to this earlier question of mine (which is unanswered :-( ) and more generally to semi-direct products of operads by bialgebras. You construction is the semi-direct product $mathtt{Com} rtimes T$ of the commutative operad $mathtt{Com}$ by $T$.

1. No, there is no diagonal in general. You need a cocommutative bimonoid object, i.e. an object equipped with a multiplication $mu : T otimes T to T$ and a comultiplication $Delta : T to T otimes T$ such that $mu$ is associative and unital, $Delta$ is coassociative, cocommutative and counital, and they satisfy a compatibility relation $Delta circ mu = mu circ (Delta otimes Delta)$.




2. I'll assume that by "free operad" you mean a left adjoint to the forgetful functor. Let $varnothing$ be the initial object of your category and suppose that $varnothing otimes X = varnothing = X otimes varnothing$ for all $X$. Then (as Simon Henry mentions in the comments), the free operad on $T$ is simply given by $mathtt{O}(1) = T$ and $mathtt{O}(n) = varnothing$ for $n neq 1$.