Irtsgbr

Let $mathcalO$ be an operad in the monoidal category $M$. Then $mathcalO(1)$ together with the morphisms
$$mathcalO(1)otimes mathcalO(1)to mathcalO(1)$$
and the unit $eta:1to mathcalO(1)$ is a monoid object. Moreover, a morphism $varphi:mathcalOto mathcalO'$ of operads induces a morphism $mathcalO(1)to mathcalO(1)$ of monoid objects. Therefore, we get a forgetful functor
$$mathrmOperads(M)to mathrmMonoids(M).$$
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
$$mathcalO_T(r) := T^otimes r.$$
and the structure maps
$$T^otimes rotimes bigotimes_i=1^r T^otimes k_ito T^otimes (k_1+dotsb+k_r)$$
as follows: Let $Delta:Tto T^otimes k$ be the diagonal (existence is clear if $otimes$ is the categorical product). Then
$$Totimes T^otimes kstackrelDeltaotimes mathrmidtoT^otimes kotimes T^otimes k cong (T^otimes 2)^otimes k to T^otimes k.$$
In $mathbfSet$, 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
   $$Tcong Totimes 1^otimes (k-1)stackrelmathrmidotimes eta^otimes (k-1)to T^otimes k.$$
   


2. Obviously $mathcalO_T(1)cong T$, but it seems to be not true that $mathcalO_T$ 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 $mathttCom rtimes T$ of the commutative operad $mathttCom$ 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 $mathttO(1) = T$ and $mathttO(n) = varnothing$ for $n neq 1$.