When you work with lax, oplax or strong monoidal functors, you want to work with bicategorical adjunctions (there will be no $1$-categorical adjoint for sure). The bijection on the Hom sets is thus replaced by an equivalence of categories.
For example, the bicategorical left adjoint of the forgetful functor $\mathsf{MonCat}_{\mathsf{strong}} \to \mathsf{Cat}$ maps a category $\mathcal{C}$ to the monoidal category $M(\mathcal{C})$ of sequences $(X_1,\dotsc,X_n)$ of objects in $\mathcal{C}$ with the obvious (in fact, strict) monoidal structure. If $(\mathcal{D},\otimes)$ is a monoidal category, then the restriction map
$$\mathrm{StrongMonFun}(M(\mathcal{C}),(\mathcal{D},\otimes)) \longrightarrow \mathrm{Fun}(\mathcal{C},\mathcal{D})$$
is an equivalence of categories, not an isomorphism of categories (or sets).
It turns out there is no bicategorical left adjoint for $\mathsf{MonCat}_{\mathsf{lax}}$. Otherwise, the image of the left adjoint at the empty category would yield a bicategorical initial object in $\mathsf{MonCat}_{\mathsf{lax}}$. This is a monoidal category $\mathcal{M}$ such that for every monoidal category $\mathcal{C}$ there is an equivalence of categories
$$\mathrm{LaxMonFun}(\mathcal{M},\mathcal{C}) \simeq \star.$$
In other words, there is a lax monoidal functor $\mathcal{M} \to \mathcal{C}$, every other is isomorphic to it, and it has only trivial monoidal endomorphisms.
But now consider the special case $\mathcal{C} = \mathcal{M}$ and the identity $\mathrm{id} : \mathcal{M} \to \mathcal{M}$ as well as the trivial lax monoidal functor $1 : \mathcal{M} \to \mathcal{M}$ which, as a functor, is the constant functor at the unit object $1$, and the lax monoidal structure consists of the evident morphisms $1 \to 1$, $1 \otimes 1 \to 1$. By assumption, we have $\mathrm{id} \cong 1$. By unpacking the definition of an isomorphism of lax monoidal functors, this implies that $\mathcal{M}$ is (as a monoidal category) equivalent to the terminal monoidal category $\{1\}$ which has just one object, one morphism, and the trivial strict monoidal structure.
However, we have an isomorphism of categories
$$\mathrm{LaxMonFun}(\{1\},\mathcal{C}) \cong \mathrm{Mon}(\mathcal{C}),$$
where $\mathrm{Mon}(\mathcal{C})$ denotes the category of monoids internal to $\mathcal{C}$. So we get
$$\mathrm{Mon}(\mathcal{C}) \simeq \star$$
for all monoidal categories $\mathcal{C}$, which is clearly absurd.
For $\mathsf{MonCat}_{\mathsf{oplax}}$ a similar argument works. If there was a bicategorical initial object, it must be $ \{1\}$ again, which together with
$$\mathrm{OplaxMonFun}(\{1\},\mathcal{C}) \cong \mathrm{CoMon}(\mathcal{C})$$
yields a contradiction.
Since there is no bicategorical initial object, there is a fortiori no initial object in the $1$-categorical sense.
