monoidality of $ A\otimes (-) $ with $ A $ monoid belonging to the center Let $(\mathcal{C}, \otimes)$ a monoidal category, and $(A, m, e)$ a monoid (where $m: A\otimes A\to A$, $e: I\to A$ ecc. ), with $(A, u)$ belonging to the centre of $(\mathcal{C}, \otimes)$: $u: A\otimes (-)\cong (-)\otimes A$, ecc. see [JS] p.38.
Consider tha (usual) functor  $F_A(-):= A\otimes (-): \mathscr{C}\to \mathscr{C}$, there are the natural morphisms: 
$\alpha_{X,Y}: F_A(X)\otimes F_A(Y)= A\otimes X\otimes A\otimes Y\xrightarrow{1u1} A\otimes A\otimes X\otimes Y
\xrightarrow{m11} A\otimes X\otimes Y=F_A(X\otimes Y)$
$\phi: I\cong I\otimes I\xrightarrow{e1}A\otimes I=F_A(I)$.
Question: Is the data above define a monoidal functor? What coherence axioms (between the monoidal and centre object structure) we need?   
[JS] Braided Tensor Categories, A.Joyal, R.Street
https://www.sciencedirect.com/science/article/pii/S0001870883710558
 A: I write $XY$ instead of $X\otimes Y$ and ignore canonical isomorphism and morphisms labels (unambiguous).
I assume the axiom:
Ax) $(AXAY\to AAXY\to AXY) = (AXAY\to XAAY\to XAY\to AXY)$.
The second member is  an alternative computation of $\alpha$.
First I have to show that:
$F_A(X)F_A(Y)F_A(Z)\to F_A(XY)F_A(Z)\to F_A(XYZ) =$
$F_A(X)F_A(Y)F_A(Z)\to F_A(X)F_A(YZ)\to F_A(XYZ)$
i.e. that
$$AXAYAZ\to AAXYAZ\to AXYAZ\to AXAYZ\to AAXYZ\to AXYZ$$
is equal to:
$$AXAYAZ\to AXAAYZ\to AXAYZ\to AAXYZ\to AXYZ$$
proceed:
$(AXAYAZ\to AAXYAZ\to AXYAZ)\to (AXAYZ\to AAXYZ\to AXYZ)=^{Ax}
=(AXAYAZ\to XAAYAZ\to XAYAZ\to AXYAZ) \to (AXAYZ\to XAAYZ\to XAYZ\to AXYZ)=^{natuality}$
$=AXAYAZ\to XAAYAZ\to (XAYAZ \to XAAYZ\to XAYZ\to AXYZ)=^{Ax}$
$=AXAYAZ\to XAAYAZ\to (XAYAZ \to XYAAZ\to XYAZ\to XAYZ\to AXYZ)=^{Ax}$
$=A_1XA_2YA_3Z\to XA_1A_2YA_3Z\to XA_{1,2}YA_3Z \to XA_{1,2}A_3YZ\to XAYZ\to AXYZ=^{naturality}$
$=A_1XA_2YA_3Z\to XA_1A_2YA_3Z\to XA_1A_2A_3YZ\to XA_{1,2}A_3YZ \to  XAYZ\to AXYZ=^{monoid}$
$=A_1XA_2YA_3Z\to XA_1A_2YA_3Z \to XA_1A_2A_3YZ\to XA_1A_{2,3}YZ \to  XAYZ\to AXYZ=^{naturality}$
$=A_1XA_2YA_3Z\to A_1XA_2A_3YZ\to XA_1A_2A_3YZ \to XA_1A_{2,3}YZ \to  XAYZ\to AXYZ=^{naturality}$
$=A_1XA_2YA_3Z\to A_1XA_2A_3YZ\to (A_1XA_{2,3}YZ\to XA_1A_{2,3}YZ \to  XAYZ\to AXYZ)=^{Ax}$
$=A_1XA_2YA_3Z\to A_1XA_2A_3YZ\to A_1XA_{2,3}YZ\to A_1A_{2,3}XYZ \to  AXYZ.$
About the unity axioms we prove that   $F_A(X)\cong IF_A(X)\to F_A(I)F_A(X)\to F(IX)\cong F(X)$ is the identity, this is::
$AX\cong IAX\to AAX\cong AIAX\cong IAAX\to IAX\cong AX$ for naturality this is:
$AX\cong IAX\to AAX\cong AIAX\cong IAAX\cong AAX\to AX$ for the usual monoidal topic this is :
$AX\cong IAX\to AAX\to AX=1$. The other unitary axiom is quite similar.
