Forgetful functor $Z(C)\rightarrow C$ has Left Adjoint

Question

The monoidal center $Z(C)$ of a monoidal category $C$ comes with a forgetful functor $F:Z(C)\rightarrow C$ defined $Z(X,\phi)=X.$ Does $F$ always admit a left adjoint? This is known (Section 3.2.) if $C$ is a tensor category. What about the general case? Tried the General Adjoint Functor Theorem but I'm not even sure if $F$ preserves limits in general.

Answer 1

Consider a nontrivial monoid $M$ with trivial center (in the sense of monoid theory, i.e. the only element of $M$ commuting with all elements of $M$ is $1\in M$). We can consider $M$ as a monoidal category $(\mathcal{M},\otimes)$ in which $\mathcal{M}$ is the discrete category on the elements of $M$, and $\otimes$ is the monoid operation of $M$. Now, since $\mathcal{M}$ only has identity morphisms and $M$ has a trivial center, the center of $(\mathcal{M},\otimes)$ is the trivial category $*$, so $F\colon *\to\mathcal{M}$ is the inclusion of the object $1$. This is a right adjoint functor iff $1$ is a terminal object of $\mathcal{M}$. Since $\mathcal{M}$ has more than one element and is discrete, this cannot be.