FALSE: the left and right internal homs agree in every monoidal category

For bimodules $A,X,Y$, a bimodule map $h:A{\otimes }_{R}X\to Y$ corresponds to the bimodule map $a⟼(x⟼h(a\otimes x))$ from $A$ to ${\mathrm{Hom}}_{-R}(X,Y)$, where $(r\phi s)(x)=r\phi (sx)$. The inverse is evaluation, $(a,x)\mapsto {\phi }_a(x)$; balancing and the two bimodule actions make both constructions well defined by [L2]. Thus ${\mathrm{Hom}}_{-R}(X,-)$ is right adjoint to $-{\otimes }_{R}X$. Similarly, maps $X{\otimes }_{R}A\to Y$ correspond to maps $A\to {\mathrm{Hom}}_{R-}(X,Y)$, with $(r\psi s)(x)=\psi (xr)s$. Hence [L1] identifies these as the right and left internal homs respectively.

A right $R$-linear map $f:R\mathrm{/}N\to R$ is determined by $a:=f(1+N)$, and it is well defined exactly when $aN=0$. Thus $[X,Y]\cong {\mathrm{ann}}_{\mathrm{\ell }}(N)=\{\begin{array}{cc}0& b\\ 0& c\end{array}:b,c\in k\}$.

A left $R$-linear map $g:R\mathrm{/}N\to R$ is determined by $b:=g(1+N)$, and it is well defined exactly when $Nb=0$. Thus $⟨X,Y⟩\cong {\mathrm{ann}}_r(N)=\{\begin{array}{cc}a& b\\ 0& 0\end{array}:a,b\in k\}$.

These two bimodules are not isomorphic: every element of ${\mathrm{ann}}_r(N)$ is annihilated on the left by $e_{22}$, while $e_{22}\cdot e_{22}=e_{22}\mathrm{\ne }0$ in ${\mathrm{ann}}_{\mathrm{\ell }}(N)$. By [L1], the left and right internal homs therefore need not agree.

Hence the statement is false.