Consequences of coherence in a monoidal category

Question

Let $\mathcal{D}_i = \left( D_i, \otimes_i, \mathbf{1}_i \right)$ be two cocomplete additive monoidal categories and let $\mathcal{F} \colon \mathcal{D}_1 \rightarrow \mathcal{D}_2$ be a strong monoidal equivalence of categories with tensor constraints isomorphisms $\mu_{X,Y} \colon \mathcal{F} \left( X \right) \otimes_2 \mathcal{F} \left( Y \right) \rightarrow \mathcal{F} \left( X \otimes_1 Y \right)$ and a unit isomorphism $\varepsilon \colon \mathbf{1}_2 \rightarrow \mathcal{F} \left( \mathbf{1}_1 \right)$.

1. Given an object $M \in D_1$, one can try and mimic the construction of the "tensor coalgebra" $T^c_{\otimes_1} \left( M \right) := \bigoplus_{i=0}^{\infty} M^{\otimes_1 i}$ together with the standard deconcatenation coproduct $\Delta \colon T^c_{\otimes_1} \left( M \right) \rightarrow T^c _{\otimes_1} \left( M \right) \otimes_1 T^c_{\otimes_1} \left( M \right)$. Formally, this involves choosing some specific bracketing for each $M^{\otimes_1 k}$ (say $M \otimes_1 \left( M \otimes_1 \left( \dots \otimes_1 \left( M \otimes_1 M \right) \right) \right)$ and then using various tensor product of associators to define the possible splittings of $m_1 \otimes_1 \left( m_2 \otimes_1 \left( \dots \left( m_{k-1} \otimes_1 m_k \right) \right) \right)$ into two parts $$\left( m_1 \otimes_1 \left( m_2 \otimes_1 \left( \dots \left( m_{i-1} \otimes_1 m_i \right) \right) \right) \right) \otimes_1 \left( m_{i+1} \otimes_1 \left( m_{i+2} \otimes_1 \left( \dots \left( m_{k-1} \otimes_1 m_k \right) \right) \right) \right) $$ and so on. I would guess that the coherence theorem for monoidal categories will somehow imply that the resulting object is indeed an "associative coalgebra" (i.e, a comonoid in $\mathcal{D}_1$) but I'm not sure how to deduce it formally. Is this true?
2. Assuming the answer to the previous question is yes, set $C = T^c \left( M \right)$ and consider $\mathcal{F} \left( C \right)$. Since $\mathcal{F}$ is strong, we can endow $\mathcal{F} \left( C \right)$ with a comonoid structure by seting $\Delta_{\mathcal{F} \left( C \right)} = \mu_{C,C}^{-1} \circ \mathcal{F} \left( \Delta_C \right)$. On the other hand, the tensor constraints isomorphisms (together with the unit isomorphism) induce an isomorphism $\Phi$ between $\mathcal{F} (C) = \mathcal{F} \left( \bigoplus_{i=0}^{\infty} M^{\otimes_1 i} \right)$ and $\bigoplus_{i=0}^{\infty} \mathcal{F} \left( M \right)^{\otimes_2 i} = T^c_{\otimes_2} \left( \mathcal{F} (M) \right)$. I would guess that the isomorphism $\Phi$ is then an isomorphism of comonoids. To verify this it seems I need to check that various diagrams involving both the associators and the tensor constraints morphisms commute. Is this true and is there some coherence theorem which guarantees that this indeed works "for free"?