Monoidal category that is not spacial Selinger ("A survery of graphical languages for monoidal categories", 2009) defines a "spacial monoidal category" as one that satisifies the following string-diagram axiom 

This diagram says that for every $h : I \to I$ and object $A$,
$$ \rho_A \circ (\text{id}_A \otimes h) \circ \rho^{-1}_A =  \lambda_A \circ (h \otimes \text{id}_A) \circ \lambda_A^{-1},$$ where $\lambda_A$ and $\rho_A$ are the left and right unitors.
What is an example of a monoidal category that is not spacial by this definition? 
 A: If your monoidal category is the fundamental 2-groupoid of a space, then this is exactly asking whether $\pi_1$ acts trivially on $\pi_2$ or not (or more precisely, that equality is saying that $A \in \pi_1$ acts trivially on $h \in \pi_2$).  If I remember right, $\mathbb{R}P^2$ gives the easiest to understand counterexample.
A: An example I learned about in Khovanov's "Heisenberg algebra and a graphical calculus", 2010, is the restriction and induction functors for the infinite chain $S_0\subset S_1\subset S_2\subset\cdots$ of symmetric groups.  This is also explained in Likeng and Savage, "Embedding Deligne's category $\operatorname{Rep}(S_t)$ in the Heisenberg category," 2019.
Let $\mathcal{S}=\prod_{m\in\mathbb{N}}\bigoplus_{n\in\mathbb{N}}\mathrm{Bim}{(S_n,S_m)}$, where each $\mathrm{Bim}(S_n,S_m)$ is the category of $(S_n,S_m)$-modules over a field $k$.  This has a monoidal structure given by tensor products of compatible bimodules, with the monoidal unit being $I=\prod_{m}k[S_m]$ with each $k[S_m]$ as an $(S_m,S_m)$-module.  We may regard $(S_n,S_m)$-modules as objects of $\mathcal{S}$ by setting all non-$m$ indices to the $0$ bimodule.
The induction functor $\mathrm{Ind}_{S_n}^{S_{n+1}}$ can be given as an object $$\mathrm{Ind}=\prod_{n\geq 1} {}_n(k[S_n])_{n-1},$$ where the subscript notation means $k[S_n]$ is treated as an $(S_n,S_{n-1})$-module.  This gives induction in the sense that, for each $S_n$-module $M$, the object $\mathrm{Ind}\otimes M$ is $\operatorname{Ind}_{S_n}^{S_{n+1}}M$.  Similarly, restriction is given by $$\mathrm{Res}=\prod_{n\geq 1}{}_{n-1}(k[S_n])_n.$$
Just as induction and restriction are biadjoint functors, the $\mathrm{Ind}$ and $\mathrm{Res}$ objects are left and right duals to each other.  Two of the four associated pairings and copairings are $\mathrm{Ind}\otimes\mathrm{Res} \to I$ and $I\to \mathrm{Ind}\otimes\mathrm{Res}$.
Since $$\mathrm{Ind}\otimes\mathrm{Res} =\prod_{n\geq 1} {}_{n}(k[S_n]\otimes_{S_{n-1}}k[S_n])_n,$$
we may give their definitions in the form
\begin{align*}
\mathrm{Ind}\otimes\mathrm{Res} &\to I\\
(g\otimes h&\mapsto gh)_{n\geq 1}\\
\end{align*}
and
\begin{align*}
I &\to \mathrm{Ind}\otimes\mathrm{Res}\\
(g& \mapsto \sum_{i=1}^n gg_i\otimes g_i^{-1})_{n\geq 1},
\end{align*}
where $g_1,\dots,g_n\in S_n$ form a set of coset representatives for $S_n/S_{n-1}$.
The composition $h:I\to \mathrm{Ind}\otimes\mathrm{Res} \to I$ of these is $h=(n\operatorname{id})_{n\geq 1}$.
We can calculate
\begin{align*}
\operatorname{id}_{\mathrm{Ind}}\otimes h &= ((n-1)\operatorname{id})_{n\geq 1}\\
h\otimes \operatorname{id}_{\mathrm{Ind}} &= (n\operatorname{id})_{n\geq 1},
\end{align*}
and therefore $\mathcal{S}$ is not spacial.
Graphically, this is that counter-clockwise loops cannot be dragged across an upward strand, imagining $\mathrm{Ind}$ as an up-arrow and $\mathrm{Res}$ as a down arrow.
A: One of the simplest examples of a non-spacial category is $\mathrm{End}(\mathrm{Vec}^{\oplus 2})$, the category of $2\times2$ matrices with vector space coefficients. Working over a field $\mathbb k$, this is a multifusion category with four simple objects:
$$ \begin{pmatrix} \mathbb k & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & \mathbb k \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ \mathbb k & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & \mathbb k \end{pmatrix}$$
and the tensor product is just matrix multiplication (with tensor product and direct sum of vector spaces, of course). The monoidal unit is not simple:
$$ \mathbb{1} = \begin{pmatrix} \mathbb k & 0 \\ 0 & 0 \end{pmatrix} \oplus \begin{pmatrix} 0 & 0 \\ 0 & \mathbb k \end{pmatrix}.$$
In particular, the endomorphisms of the monoidal unit is the commutative ring $\mathbb k \oplus \mathbb k$ (with componentwise addition and multiplication). Consider the endomorphism
$$ h = (\alpha,\beta) \in \mathrm{End}(\mathbb 1)$$
and object
$$ A = \begin{pmatrix} 0 & \mathbb k \\ 0 & 0 \end{pmatrix}.$$
Then one of your diagrams evaluates to $\alpha \, \mathrm{id}_A$ and the other one evaluates to $\beta \, \mathrm{id}_A$.
Remark: Write $R = \mathrm{End}(\mathbb 1) = \mathbb k \oplus \mathbb k$. Then it is an associative $\mathbb k$-algebra, and the multifusion category above is the monoidal category of $R$-$R$ bimodules. Categories of bimodules are typically not spacial except when the algebra has trivial (i.e. one-dimensional) centre. A fun example to think through is to consider $\mathbb C$ as an $\mathbb R$-algebra. Then the category of $\mathbb C$-$\mathbb C$ bimodules has two simple objects (Exercise: why?), and is not spacial (Exercise: why not?).
